On the Existence of\\ Hermitian Self-Dual\\ Extended Abelian Group Codes
Lilibeth Dicuangco\footnote{Mathematics Department
Diliman
1101 Philippines
Vivatsgasse 7
D-53111 Bonn
Germany
I3S ESSI
BP 145
Route des Colles
France
Split group codes are a class of group algebra codes over an abelian group. They were introduced by Ding, %%@ Kohel and Ling in \cite{cunsheng} as a generalization of the cyclic duadic codes. For a prime power $q$ %%@ and an abelian group $G$ of order $n$ such that gcd$(n,q)=1$, consider the group algebra %%@ $\FF_{q^{2}}[G^{*}]$ of $\FF_{q^{2}}$ over the dual group $G^{*}$ of $G$. We prove that every ideal code %%@ in $\FF_{q^{2}}[G^{*}]$ whose extended code is Hermitian self-dual is a split group code. We characterize %%@ the orders of finite abelian groups $G$ for which an ideal code of $\FF_{q^{2}}[G^{*}]$ whose extension is %%@ Hermitian self-dual exists and derive asymptotic estimates for the number of non-isomorphic abelian groups %%@ with this property.
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\begin{document} 

\title{On the Existence of\\ Hermitian Self-Dual\\ Extended Abelian Group Codes}

\author{Lilibeth Dicuangco\footnote{Mathematics Department, University of the Philippines, Diliman, Quezon %%@
City, 1101 Philippines, {\tt ldicuangco@math.upd.edu.ph}}, \: \setcounter{footnote}{6} Pieter %%@
Moree\footnote{Max-Planck-Institut, Vivatsgasse 7, D-53111 Bonn, Germany, {\tt moree@mpim-bonn.mpg.de}},  %%@
\: \setcounter{footnote}{3} Patrick Sol\'e\footnote{CNRS, I3S ESSI, BP 145, Route des Colles, 06 903 %%@
Sophia Antipolis, France, {\tt sole@essi.fr}}}

\date{} 
\maketitle 
\date{} 
\begin{abstract}
Split group codes are a class of group algebra codes over an abelian group. They were introduced by Ding, %%@
Kohel and Ling in \cite{cunsheng} as a generalization of the cyclic duadic codes. For a prime power $q$ %%@
and an abelian group $G$ of order $n$ such that gcd$(n,q)=1$, consider the group algebra %%@
$\FF_{q^{2}}[G^{*}]$ of $\FF_{q^{2}}$ over the dual group $G^{*}$ of $G$. We prove that every ideal code %%@
in $\FF_{q^{2}}[G^{*}]$ whose extended code is Hermitian self-dual is a split group code. We characterize %%@
the orders of finite abelian groups $G$ for which an ideal code of $\FF_{q^{2}}[G^{*}]$ whose extension is %%@
Hermitian self-dual exists and 
derive asymptotic estimates for the number of non-isomorphic abelian groups %%@
with this property.
\end{abstract}

\noindent {\bf Keywords}:  counting function, extended group codes, group algebra codes, Hermitian %%@
self-dual codes, non-isomorphic abelian groups, split group codes, splittings.

\noindent {\bf Mathematics Subject Classification}: 11N64, 94B05, 11N37

\section{Introduction}

Binary duadic codes were first introduced in 1984 by Leon, Masley and Pless \cite{leon et al} as a %%@
generalization of quadratic residue codes. Smid \cite{smidthesis} generalized these results further by %%@
defining duadic codes over arbitrary finite fields in terms of a \emph{splitting} of the length of the %%@
code.  The Q-codes of Pless \cite{qcodes} are then duadic codes over $\FF_{4}$ in this setting.

Quadratic residue codes have also been generalized in a different direction (i.e., see \cite{ward}). In %%@
this approach, quadratic residue codes are defined as ideals of abelian group algebras, which is a  %%@
generalization of cyclic codes. Rushanan \cite{rushanan} proceeded to define duadic codes in this setting.

In \cite{cunsheng}, Ding, Kohel and Ling defined split group codes as ideals of abelian group algebras. %%@
Their construction makes use of a \emph{splitting} of the abelian group. Under this definition, split %%@
group codes are seen as a generalization of duadic codes.

In this paper, we consider the finite field $F = \FF_{q^2}$ and an abelian group $G$ of order $n$ such %%@
that gcd$(n,q)=1$. Following the treatment in \cite{cunsheng}, we work with the dual group $G^{\ast}$ of %%@
$G$ and consider the group algebra $F\left[ G^{\ast} \right]$. We prove 
that every ideal code in $\FG$ whose extension by a suitable parity-check is Hermitian self-dual is a %%@
split group code (Corollary \ref{Cor exte2}).
We then give sufficient and necessary conditions on the order of the group $G$ for the existence of %%@
Hermitian self-dual extended  ideal codes (Theorem \ref{Thm concon2}). We conclude the
paper by deriving asymptotic estimates on $HSD(x)$, the number of non-isomorphic abelian groups of order %%@
less than $x$ for which a Hermitian self-dual extended ideal code exists (Theorem \ref{twee}).


\section{Preliminaries}

All of the results in this section are taken from \cite{cunsheng}. In general, these results work for any %%@
finite field but since we will be dealing with Hermitian duality, we restrict our study to finite fields %%@
of square order.

Let $R$ be a finite commutative ring with unity. Let $G$ be its underlying finite abelian group written %%@
additively. Denote the order and exponent of $G$ by $n$ and $m$, respectively. Let $q$ be a power of a %%@
prime $p_{1}$ such that gcd$(n,q) = 1$ or equivalently, gcd$(m,q) = 1$. Let $F = \FF_{q^{2}}$. Let $K$ be %%@
the smallest extension of $F$ containing all the $m$-th roots of unity.
 
Let $G^{\ast}$ be the set of all characters of $G$ into $K$. The groups $G$ and $\G$ are isomorphic. Let %%@
$K \left[ G^{\ast} \right]$ be the group algebra of $K$ over $G^{\ast}$. The elements of $\KG$ are the %%@
sums $\sum_{\psi \in \G} a_{\psi}\psi$ where the $a_{\psi}$'s are elements of $K$. An ideal $I$ of $\KG$ %%@
is called an \emph{ideal code}.

The dimension of the commutative group algebra $\KG$ over $K$ is $n$. This group algebra $\KG$ contains a %%@
subgroup isomorphic to $G^{\ast}$. For any character $\psi$ in $\G$, we also denote by $\psi$ the %%@
corresponding element in $\KG$. 

If $x \in G$ and $f = \sum_{\psi \in \G} a_{\psi}\psi \in \KG$, define $f(x) = \sum_{\psi \in \G} %%@
a_{\psi}\psi(x)$.  Thus we can view the elements of $\KG$ as functions from $G$ to $K$.


\subsection{Ideal codes and Idempotent Generators}

An element $e$ of a ring is called an idempotent if $e^{2} = e$. An idempotent is called primitive if for %%@
every other idempotent $f$, either $ef = e$ or $ef = 0$.


\begin{prop} {\rm (\cite{cunsheng}.)} The primitive idempotents of $\KG$ are the elements
\begin{equation*}
e_{x} = \frac{1}{n} \sum_{\psi \in \G} \psi(x)^{-1}\psi,
\end{equation*}
for each element $x$ of $G$.
\end{prop}


\begin{prop} {\rm (\cite{cunsheng}.)} The ring $\KG$ decomposes as a direct sum $\oplus_{x \in G} K %%@
e_{x}$. If $f \in \KG$, then $f$ has the form
\begin{equation*}
f = \sum_{x \in G} f(x) e_{x}.
\end{equation*}
Every idempotent $e$ in $\KG$ can be uniquely written in the form
\begin{equation*}
e = \sum_{x \in X} e_{x},
\end{equation*}
for some non-empty subset $X$ of $G$.
\end{prop}


\noindent Let $X$ be a non-empty subset of $G$ and define the ideal
\begin{equation*}
I_{X} = \{ f \in \KG \mid f(x) = 0 \; \mbox{ for all } x \in X \}.
\end{equation*}

\begin{cor} {\rm (\cite{cunsheng}.)}
For every ideal $I$ in $\KG$, there is a unique proper subset $X$ of $G$ such that $I = I_{X}$ and $I$ is %%@
generated by the idempotent $e = \sum_{x \notin X} e_{x}$.
\end{cor}


\subsection{Split Group Codes}



Let $s$ be an element of $R$. Consider the endomorphism of $G$ given by $\tau_{s} : x \longrightarrow sx$. %%@
This induces a map $\mu_{s}$ on $\G$  given by $\mu_{s}(\psi) = \psi\circ\tau_{s}$ for each element $\psi$ %%@
of $\G$. This extends to a map on $\KG$, also denoted by $\mu_{s}$, defined by $\mu_{s}(f) = %%@
f\circ\tau_{s}$ for all $f \in \KG$. That is, for $f \in \KG$, $\mu_{s}(f)(x) = f(sx)$ for every $x \in %%@
G$.

A \emph{splitting} of $G$ over $Z$ is a triple $(Z,X_{0},X_{1})$ which gives a partition $G = Z \cup X_{0} %%@
\cup X_{1}$ such that there exists an invertible element $s$ of $R$ with $\tau_{s}(X_{0}) = X_{1}$ and %%@
$\tau_{s}(X_{1}) = X_{0}$. Under these conditions, $s$ is said to \emph{split} the triple %%@
$(Z,X_{0},X_{1})$. In addition, an invertible element $r$ of $R$ is said to \emph{stabilize} the splitting %%@
if $\tau_{r}(X_{0}) = X_{0}$ and $\tau_{r}(X_{1}) = X_{1}$.


Given a splitting $(Z,X_{0},X_{1})$, let $C_{0}(K)$ be the ideal $I_{X_{0}}$ over $K$ and let $C_{1}(K)$ %%@
be the ideal $I_{X_{1}}$ over $K$. The ideal $C_{0}(K)$ is defined as the \emph{split group code} %%@
associated to the splitting, and the ideal $C_{1}(K)$ is called the \emph{conjugate split group code}. The %%@
following notations are used to denote some special subcodes: $C_{0}^{Z}(K) = I_{Z \cup X_{0}}$,  %%@
$C_{1}^{Z}(K) = I_{Z \cup X_{1}}$ and $C_{Z}(K) = I_{X_{0} \cup X_{1}}$.

Let $s$ be an invertible element of $R$. The element $s$ is said to \emph{split} the group code $C_{0}(K)$ %%@
if $\mu_{s}(C_{0}(K)) = C_{1}(K)$ and $\mu_{s}(C_{1}(K)) = C_{0}(K)$, while $s$ is said to %%@
\emph{stabilize} the code $C_{0}(K)$ if $\mu_{s}(C_{0}(K)) = C_{0}(K)$ and $\mu_{s}(C_{1}(K)) = C_{1}(K)$.

\begin{prop} {\rm (\cite{cunsheng}.)}
Let $s$ be a unit in $R$. A split group code $C_{0}(K)$ is split or stabilized by $s$ if and only if $s$ %%@
splits or stabilizes $(Z,X_{0},X_{1})$, respectively.
\label{s}
\end{prop}


\begin{thm} {\rm (\cite{cunsheng}.)}
Let $(Z,X_{0},X_{1})$ be a splitting. Let $C_{0}(K)$ be the split group code associated to this splitting. %%@
Then the following hold:
\begin{enumerate}
\item The codes $C_{0}(K)$ and $C_{1}(K)$ are generated by the idempotents
\begin{equation*}
e = \sum_{x \notin X_{0}} e_{x} \; \mbox{ and } \; f = \sum_{x \notin X_{1}} e_{x}.
\end{equation*}
The codes $C_{0}^{Z}(K)$, $C_{1}^{Z}(K)$ and $C_{Z}(K)$ are generated by 
\begin{equation*}
\sum_{x \in X_{1}} e_{x}, \; \sum_{x \in X_{0}} e_{x}, \; \mbox{ and } \; \sum_{z \in Z} e_{z}.
\end{equation*}
\item If the splitting is given by $s$, then $\mu_{s}$ induces an equivalence of $C_{0}(K)$ with its %%@
conjugate $C_{1}(K)$, and of the subcode $C_{0}^{Z}(K)$ with $C_{1}^{Z}(K)$.
\item $\KG$ decomposes as a direct sum $C_{Z}(K) \oplus C_{0}^{Z}(K) \oplus C_{1}^{Z}(K)$.
\end{enumerate}
\label{equi}
\end{thm}


\begin{cor} {\rm (\cite{cunsheng}.)}
The codes $C_{0}(K)$ and $C_{1}(K)$ have dimension $(n + |Z|)/2$. The subcodes $C_{0}^{Z}(K)$ and %%@
$C_{1}^{Z}(K)$ have dimension $(n - |Z|)/2$. The subcode $C_{Z}(K)$ has dimension $|Z|$.
\end{cor}


\subsection{Split Group Codes Over $\FF_{q^{2}}$}

In the previous section, the split group codes are defined over the field $K$ which is assumed to contain %%@
all the $m$-th roots of unity. In this paper, we want our split group codes to be defined over the %%@
subfield $F =\FF_{q^{2}}$ without requiring $\FF_{q^{2}}$ to contain any $m$-th roots of unity.
In this section, we present sufficient and necessary conditions given in \cite{cunsheng} for split group %%@
codes to be defined over a subfield of $K$.

Let $V = V(K)$ be a vector subspace of $K^{n} = \KG$. Define $V(F) = V(K) \cap F^{n}$. Clearly, %%@
$\mbox{dim}_{F}(V(F)) \leq \mbox{dim}_{K}(V(K))$. If equality holds then we say that $V$ is \emph{defined %%@
over the field $F$}.

If the vector subspace $C_{0}(K)$ of $K^{n} = \KG$ is defined over $F$, we simply write $C_{0}$ for the %%@
subcode $C_{0}(F) = C_{0}(K) \cap F^{n}$ in $F^{n} = \FG$. In this case, we call $C_{0}$ \emph{the split %%@
group code over $F$}. Similarly, we write $C_{1}$, $C_{0}^{Z}$, $C_{1}^{Z}$ and $C_{Z}$ for the other %%@
codes defined over $F$.

Note that $(m,q)=1$ by assumption, so the integer $-q$ as an element of the finite ring $R$ is invertible %%@
and $\tau_{q^2}$ is a well-defined automorphism of $G$. The action of the group generated by %%@
$\tau_{q^{2}}$ on the elements of $G$ partitions $G$ into disjoint orbits. These $\langle \tau_{q^{2}} %%@
\rangle$-orbits play the same role as the cyclotomic cosets for the cyclic codes.


\begin{prop} {\rm (\cite{cunsheng}.)}
The idempotents of $\FG$ are those $e$ in $\KG$ of the form
\begin{equation*}
e = \sum_{x \in Y} e_{x},
\end{equation*}
where $Y$ is a union of $\langle \tau_{q^{2}} \rangle$-orbits in $G$. An idempotent $e$ in $\FG$ is %%@
primitive if and only if $Y = \langle \tau_{q^{2}} \rangle x$ for some $x \in G$.
\end{prop}

\begin{cor} {\rm (\cite{cunsheng}.)}
Let $\{ e_{1}, e_{2}, \ldots, e_{r} \}$ be the set of all primitive idempotents of $\FG$. Then every %%@
nonzero ideal $I$ of $\FG$ is generated by $e = \sum_{i \in T} e_{i}$  where $T$ is a non-empty subset of %%@
$\{1, 2,\ldots, r \}$.
\end{cor}



\begin{thm} {\rm (\cite{cunsheng}.)}
Let $I$ be an ideal in $\KG$. Then the following conditions are equivalent.
\begin{enumerate}
\item The ideal $I$ is defined over $F$.
\item The set $X = \{ x \in G \: | \: f(x) = 0 \; \mbox{ for all } f \in I  \}$ is a union of $\langle %%@
\tau_{q^{2}} \rangle$-orbits.
\item The idempotent generator of $I$ lies in $\FG$. 
\end{enumerate}
\label{descent}
\end{thm}

\begin{cor} {\rm (\cite{cunsheng}.)}
If $C_{0}(K)$ is defined over $F$ then so is $C_{1}(K)$. Moreover $\FG$ has the decomposition
\begin{equation*}
\FG = C_{Z}(F) \oplus C_{0}^{Z}(F) \oplus C_{1}^{Z}(F). 
\end{equation*}
If $s$ gives the splitting, then $\mu_{s}$ gives an equivalence of $C_{0}$ and $C_{1}$ and of  $C_{0}^{Z}$ %%@
and $C_{1}^{Z}$.
\end{cor}


\begin{thm} {\rm (\cite{cunsheng}.)}
Let $C$ be a code in $\KG$.  The block length, dimension, and minimum distance are well-defined invariants %%@
of $C$, independent of the field over which $C$ is defined.
\end{thm}


\section{Hermitian Duality and Extended Ideal Codes in $\FG$}

In this section, we present results concerning the Hermitian orthogonality of ideal codes in $\FG$. We %%@
prove that every Hermitian self-orthogonal ideal code in $\FG$ is a subcode of a split group code for some %%@
splitting of $G$ given by $-q$. We then proceed to define an extension of  
ideal codes
and determine conditions for the extended code to be Hermitian self-dual. We also give necessary and %%@
sufficient conditions on the order of the abelian group $G$ for the existence of Hermitian self-dual %%@
extended ideal codes.


\subsection{Hermitian Orthogonality of Ideal Codes in $\FG$}

Let $f = \sum_{\psi \in \G} a_{\psi} \psi$ and $g = \sum_{\psi \in \G} b_{\psi} \psi$ be elements of %%@
$\FG$. The \emph{Hermitian inner product} between $f$ and $g$ is defined as $\langle f, g \rangle_{H} = %%@
\sum_{\psi \in \G} a_{\psi}b_{\psi}^{q}$.

Let $C$ be a code in $\FG$. The Hermitian dual of $C$ is the ideal $C^{\bot_{H}} = \{ f \in \FG \; | \; %%@
\langle f, g \rangle_{H} = 0 \; \mbox{ for all } g \in C \}$. The code $C$ is said to be \emph{Hermitian %%@
self-orthogonal} if $C \subseteq C^{\bot_{H}}$ and is \emph{Hermitian self-dual} if $C = C^{\bot_{H}}$.

Theorem \ref{Thm HermSplit} states the main result of this section. It is a generalization of Proposition %%@
4.4 in \cite{DMS} to split group codes. We first prove some basic results concerning the Hermitian duals %%@
of split group codes.


\begin{prop}
Let $f = \sum_{\psi \in \G} a_{\psi} \psi$ and $g = \sum_{\psi \in \G} b_{\psi} \psi$ be elements of %%@
$\FG$. Then the Hermitian inner product of $f$ and $g$ is
\begin{equation*}
\langle f, \: g \rangle_{H} = \frac{1}{n}\sum_{x \in G}f(x)g(-q^{-1}x)^{q}
\end{equation*}
\label{hermdot}
\end{prop}
{\it Proof}. Note that 
$\mu_{-q^{-1}}(g)(x) = g(-q^{-1}x)  =  \sum_{\psi \in \G}b_{\psi}\psi(-q^{-1}x)
                         =  \sum_{\psi \in \G}b_{\psi} \psi(q^{-1}x)^{-1}.$
Thus
 $(\mu_{-q^{-1}}(g)(x))^{q}  =  \sum_{\psi \in \G}b_{\psi}^{q} \psi(q^{-1}x)^{-q}
 						    =  \sum_{\psi \in \G}b_{\psi}^{q} \psi(x)^{-1},$			
or equivalently, $(\mu_{-q^{-1}}(g))^{q} = \sum_{\psi \in \G}b_{\psi}^{q} \psi^{-1}$.


Define $f \ast g = f(\mu_{-q^{-1}}(g))^{q}$.
Then the coefficient of the trivial character of $f \ast g$ is $\sum_{\psi \in \G} a_{\psi}b_{\psi}^{q}$, %%@
which is $\langle f, g \rangle_{H}$.
Expanding $f \ast g$ in terms of its idempotent decomposition, we get
\begin{eqnarray*}
 f \ast g & = & \sum_{x \in G}(f(\mu_{-q^{-1}}(g))^{q})(x)e_{x} \\
          & = & \sum_{x \in G}f(x)(\mu_{-q^{-1}}(g)(x))^{q}e_{x} \\
          & = & \sum_{x \in G}f(x)(g(-q^{-1}x))^{q}e_{x} \\
          & = & \frac{1}{n} \sum_{\psi \in \G} \sum_{x \in G} f(x)(g(-q^{-1}x))^{q} \psi^{-1}(x)\psi.
\end{eqnarray*}
Using this expansion, the coefficient of the trivial character of $f \ast g$ is $\frac{1}{n} \sum_{x \in %%@
G}f(x)(g(-q^{-1}x))^{q}$. The result follows.
\hfill \Q{\Box}


\begin{prop}
Let $C$ be an ideal in $\KG$ which is defined over $F$. Suppose $C = I_{X}$ for some non-empty subset $X$ %%@
of $G$. Then $C^{\bot_{H}} = I_{X'}$ where $X' = G \setminus \tau_{-q}(X)$.
\label{hermdual}
\end{prop}
{\it Proof}. {}From Theorem \ref{descent}, $X$ is a union of $\langle \tau_{q^{2}} \rangle$-orbits.
Let $X' = G \setminus \tau_{-q}(X)$. Let $f \in I_{X'}$ and $g \in C = I_{X}$. By Proposition %%@
\ref{hermdot}, $\langle f, g \rangle_{H} = \frac{1}{n}\sum_{x \in G}f(x)(g(-q^{-1}x))^{q}$.  Since $X$ and %%@
$\tau_{-q}(X)$  are unions of $\langle \tau_{q^{2}} \rangle$-orbits and clearly $x$ and $q^{2}x$ belong to %%@
the same $\tau_{q^{2}}$-orbit, it follows that
$-q^{-1}x \in X$ if and only if $(-q)^{2}(-q^{-1})x \in X$ 
                 if and only if $-qx \in X$
                 if and only if $(-q)(-q)x \in \tau_{-q}(X)$
				 if and only if  $x \in \tau_{-q}(X)$. 
Thus $g(-q^{-1}x) = 0$ for all $x \in \tau_{-q}(X)$. Since $f \in I_{X'}$, $f(x) = 0$ for every $x \in X' %%@
= G \setminus \tau_{-q}(X)$. Therefore $\sum_{x \in G} f(x)(g(-q^{-1}x))^{q} = 0$, implying that $\langle %%@
f, g \rangle_{H} = 0$. Thus $I_{X'} \subseteq C^{\bot_{H}}$. Comparing dimensions, we get $C^{\bot_{H}} = %%@
I_{X'}$.
\hfill \Q{\Box}

\vskip .1in
\noindent {\tt Remark}. We note that for any subset $X$ of $G$ which is a union of $\langle \tau_{q^{2}} %%@
\rangle$-orbits of $G$, we have $\tau_{-q^{-1}}(X) = \tau_{(-q)^{2}}(\tau_{-q^{-1}}(X)) = \tau_{-q}(X)$.


\begin{prop}
Let $(Z,X_{0},X_{1})$ be a splitting of $G$ over $Z$ where $Z$, $X_{0}$ and $X_{1}$ are unions of $\langle %%@
\tau_{q^{2}} \rangle$-orbits. The ring element $-q$ splits or stabilizes $C_{0}$ if and only if $-q^{-1}$ %%@
splits or stabilizes $C_{0}$, respectively.
\label{s2}
\end{prop}
{\it Proof}. Using Proposition \ref{s}, we need only show that $-q$ splits or stabilizes $(Z,X_{0},X_{1})$ %%@
if and only if  $-q^{-1}$  splits or stabilizes $(Z,X_{0},X_{1})$, respectively. {}From the remark above, %%@
$\tau_{-q^{-1}}(X_{0}) = \tau_{-q}(X_{0})$ and $\tau_{-q^{-1}}(X_{1}) = \tau_{-q}(X_{1})$. The result %%@
follows.
\hfill \Q{\Box}


\begin{prop}
Let $(Z,X_{0},X_{1})$ be a splitting of $G$ over $Z$. Assume that $Z$, $X_{0}$ and $X_{1}$ are unions of %%@
$\langle \tau_{q^{2}} \rangle$-orbits.  Suppose $Z$ is stabilized by $\tau_{-q}$. Then $C_{Z}^{\bot_{H}} = %%@
C_{0}^{Z} \oplus C_{1}^{Z}$. If $-q$ splits $C_{0}$, then $C_{0}^{\bot_{H}} = C_{0}^{Z}$. If $-q$ %%@
stabilizes $C_{0}$, then $C_{0}^{\bot_{H}} = C_{1}^{Z}$.
\label{OddSplit}
\end{prop}
{\it Proof}. Note that $\tau_{-q^{-1}}(Z) = \tau_{(-q)^{2}}(\tau_{-q^{-1}}(Z)) = \tau_{-q}(Z)$ and by %%@
assumption, $\tau_{-q}(Z) = Z$.
Let $f \in C_{Z} = I _{X_{0} \cup X_{1}}$ and let $g \in C_{0}^{Z} \oplus C_{1}^{Z}$. Then
\begin{equation*}
\langle f, g \rangle_{H} = \frac{1}{n} \sum_{x \in G}f(x)(g(-q^{-1}x))^{q} = 0,
\end{equation*}
since $f(x) = 0$ for all $x \in X_{0} \cup X_{1}$ and $g(-q^{-1}x) = 0$ for all $x \in Z$. Thus $C_{0}^{Z} %%@
\oplus C_{1}^{Z} \subseteq C_{Z}^{\bot_{H}}$. Comparing dimensions, we have $C_{Z}^{\bot_{H}} = C_{0}^{Z} %%@
\oplus C_{1}^{Z}$.

Suppose $-q$ splits $C_{0}$. Let $f \in C_{0}$ and $g \in C_{1}^{Z}$. Note that $\tau_{-q^{-1}}(Z) = Z$. %%@
By assumption, $\tau_{-q}(X_{1}) = X_{0}$, or equivalently, $\tau_{-q^{-1}}(X_{1}) = X_{0}$. Clearly $f(x) %%@
= 0$ for all $x \in X_{0}$ and $g(-q^{-1}x) = 0$ for all $x \in Z \cup X_{1}$. Thus 
\begin{equation*}
\langle f, g \rangle_{H} = \frac{1}{n} \sum_{x \in G}f(x)(g(-q^{-1}x))^{q} = 0,
\end{equation*}
implying that $C_{0}^{Z} \subseteq C_{0}^{\bot_{H}}$. Comparing dimensions, we have $C_{0}^{\bot_{H}} = %%@
C_{0}^{Z}$.


Suppose $-q$ stabilizes $C_{0}$. Then $\tau_{-q^{-1}}(X_{1}) = \tau_{-q}(X_{1}) = X_{1}$. Let $f \in %%@
C_{0}$ and $g \in C_{1}^{Z}$. Clearly $f(x) = 0$ for all $x \in X_{0}$ and  $g(-q^{-1}x) = 0$ for all $x %%@
\in Z \cup X_{1}$. Thus 
\begin{equation*}
\langle f, g \rangle_{H} = \frac{1}{n} \sum_{x \in G}f(x)(g(-q^{-1}x))^{q} = 0,
\end{equation*}
implying that $C_{1}^{Z} \subseteq C_{0}^{\bot_{H}}$. Comparing dimensions, we have $C_{1}^{\bot_{H}} = %%@
C_{0}^{Z}$.
\hfill \Q{\Box}

\vskip .2in

\noindent We now prove the main result of this section.


\begin{thm}
Let $C = I_{X}$ be an ideal in $\FG$. Then $C$ is Hermitian self-orthogonal if and only if $C = C_{0}^{Z}$ %%@
for some splitting $(Z, X_{0}, X_{1})$ of $G$ which is split by $-q$ (that is, $C$ is a subcode of a split %%@
group code which is split by $-q$).
\label{Thm HermSplit}
\end{thm}
{\it Proof}.

\noindent $(\Longleftarrow)$ Suppose $(Z, X_{0}, X_{1})$ is a splitting of $G$ which is split by $-q$. Let %%@
$C = C_{0}^{Z}$. By Proposition \ref{OddSplit}, $C^{\bot_{H}} = (C_{0}^{Z})^{\bot_{H}} = C_{0} \supseteq %%@
C_{0}^{Z} = C$.


\noindent $(\Longrightarrow)$ Let $C^{\bot_{H}} = I_{X'}$. By Proposition \ref{hermdual}, $X' = G %%@
\setminus \tau_{-q}(X)$. By assumption, $C \subseteq C^{\bot_{H}}$. Thus $X' \subseteq X$, or $G \setminus %%@
\tau_{-q}(X) \subseteq X$. Note that both $X$ and $X'$ are unions of $\langle \tau_{q^{2}} %%@
\rangle$-orbits. Write $X = Z \cup X_{0}$ with
$\tau_{-q}(Z) = Z \; \mbox{ and } \; Z \cap X_{0} = \emptyset$.
Since $\tau_{-q}$ either fixes a $\langle \tau_{q^{2}} \rangle$-orbit or it sends it to another $\langle %%@
\tau_{q^{2}} \rangle$-orbit, we are actually choosing $Z$ as the union of $\langle \tau_{q^{2}} %%@
\rangle$-orbits contained in $X$ which are fixed by $\tau_{-q}$ and $X_{0}$ as the complement of $Z$ in %%@
$X$, so that clearly $\tau_{-q}(X_{0}) \cap X_{0} = \emptyset$.


We first show that neither $Z$ nor $X_{0}$ is empty. If $0 \notin X$ then $0 \notin \tau_{-q}(X)$, which %%@
implies that $0 \in G \setminus \tau_{-q}X$. But $G \setminus \tau_{-q}(X) \subseteq X$, implying that $0 %%@
\in X$, a contradiction. Thus $0 \in X$, and so by our choice of partition of $X$, $0 \in Z$ proving that %%@
$Z \not= \emptyset$. If $X_{0} = \emptyset$ then $X = Z$ and $\tau_{-q}(X) = X$. Thus $X' = G \setminus %%@
\tau_{-q}(X) = G \setminus X$ which implies that $X' \cap X = \emptyset$, a contradiction since $X' %%@
\subseteq X$. Thus $X_{0} \not= \emptyset$.

Note that $X' = G \setminus \tau_{-q}(X) = G \setminus Z \cup \tau_{-q}(X_{0})$. Since $X_{0} \cap Z = %%@
X_{0} \cap \tau_{-q}(X_{0}) = \emptyset$, we have $X_{0} \subseteq X'$. Consider $\tau_{-q}(X') = G %%@
\setminus X$. Since $X' \subseteq X$, it follows that $\tau_{-q}(X') \cap X' = \emptyset$
and $\tau_{-q}$ does not fix any $\langle \tau_{q^{2}} \rangle$-orbit contained in $X'$, implying that $X' %%@
\subseteq X_{0}$. Hence $X' = X_{0}$.
Let $X_{1} = \tau_{-q}(X')$. Then $(Z, X_{0}, X_{1})$ gives a splitting of $G$ such that $\tau_{-q}(Z) = %%@
Z$, $\tau_{-q}(X_{0}) = X_{1}$ and $\tau_{-q}(X_{1}) = X_{0}$  and $C = I_{X} = I_{Z \cup X_{0}} = %%@
C_{0}^{Z}$.
\hfill \Q{\Box}


\subsection{Extensions of Ideal Codes in $\FG$}

Using the terminology of split group codes, duadic codes are easily seen to be split group codes for %%@
splittings of $G$ over $Z = \{0 \}$ where $G$ is cyclic (see Example III.1 of \cite{cunsheng}). In %%@
\cite{DMS}, we defined an extension for an odd-like duadic code and gave a sufficient condition for the %%@
extended code to be Hermitian self-dual. In this section, we consider split group codes for the abelian %%@
group $G$ with splittings over $Z = \{0 \}$ and derive analogous results regarding Hermitian self-duality %%@
of the extended split group codes.

Let the order $n$ of the abelian group $G$ be odd. Consider the equation
\begin{equation}
\frac{1}{n} + \gamma^{q+1} = 0,
\label{gam}
\end{equation}
which is solvable in $\FF_{q^{2}}$.
Let $\gamma$ be a solution to (\ref{gam}). For each $f \in \FG$, define $\widetilde{f} = (f, -\gamma f(0)) %%@
\in \FG \times F$. If $C$ is a code in $\FG$ then the extended code $\widetilde{C}$ is defined as the %%@
subspace
\begin{equation*}
\widetilde{C} = \{ \widetilde{f} = (f, -\gamma f(0)) \mid f \in C \} \subseteq \FG \times F.
\end{equation*}
 
 
 \begin{prop}
 Let $(Z = \{0 \}, X_{0}, X_{1})$ be a splitting of $G$ where $Z$, $X_{0}$ and $X_{1}$ are unions of %%@
$\langle \tau_{q^{2}} \rangle$-orbits. Let  $C_{0}$ be the corresponding split group code defined over $F %%@
= \FF_{q^{2}}$.
\begin{enumerate}
\item The extended codes $\widetilde{C_{0}}$ and $\widetilde{C_{1}}$ are equivalent. 
\item If $-q$ splits $C_{0}$, then $\widetilde{C_{0}}^{\bot_{H}} = \widetilde{C_{0}}$ and %%@
$\widetilde{C_{1}}^{\bot_{H}} = \widetilde{C_{1}}$.
\item If $-q$ stabilizes $C_{0}$, then $\widetilde{C_{0}}^{\bot_{H}} = \widetilde{C_{1}}$ and %%@
$\widetilde{C_{1}}^{\bot_{H}} = \widetilde{C_{0}}$.
\end{enumerate}
\label{Thm exte2}
 \end{prop}
{\it Proof}. The equivalence of $\widetilde{C_{0}}$ and $\widetilde{C_{1}}$ is an immediate consequence of %%@
Theorem \ref{equi}.
 
 
Suppose $-q$ splits $C_{0}$. Let $f$, $g$ be elements of $C_{0}$. {}From Proposition \ref{s2}, $-q^{-1}$ %%@
also splits $C_{0}$. Thus $\mu_{-q^{-1}}(g) \in C_{1}$. Note that $-q^{-1}x \in X_{0} \iff x \in %%@
\tau_{-q^{-1}}(X_{0}) = X_{1}$. It follows that $f(x) = 0$ for all $x \in X_{0}$ and $g(-q^{-1}x) = 0$ for %%@
all $x \in X_{1}$. Hence using Proposition \ref{hermdot}, we get
\begin{eqnarray*}
\langle f, g \rangle_{H} & = & \frac{1}{n} f(0) g(0)^{q} = -\gamma^{q+1} f(0) g(0)^{q}.
\end{eqnarray*}
Thus $\langle \widetilde{f}, \widetilde{g} \rangle_{H} = 0$ and $\widetilde{C_{0}}^{\bot_{H}} = %%@
\widetilde{C_{0}}$. By a similar argument, it can be shown that $\widetilde{C_{1}}^{\bot_{H}} = %%@
\widetilde{C_{1}}$.

Suppose $-q$ stabilizes $C_{0}$. Let $f \in C_{0}$ and let $g \in C_{1}$. The element $-q^{-1}$ also %%@
stabilizes $C_{0}$ and $\mu_{-q^{-1}}(g) \in C_{0}$. Again, $-q^{-1}x \in X_{1} \iff x \in %%@
\tau_{-q^{-1}}(X_{1}) = X_{1}$. So $f(x) = 0 $ for all $x \in X_{0}$ and $g(-q^{-1}x) = 0$ for all $x \in %%@
X_{1}$. Thus
\begin{eqnarray*}
\langle f, g \rangle_{H} & = & \frac{1}{n} f(0) g(0)^{q} = -\gamma^{q+1} f(0) g(0)^{q},
\end{eqnarray*}
and so $\langle \widetilde{f}, \widetilde{g} \rangle_{H} = 0$ and $\widetilde{C_{0}}^{\bot_{H}} = %%@
\widetilde{C_{1}}$. Similarly, $\widetilde{C_{1}}^{\bot_{H}} = \widetilde{C_{0}}$.
\hfill \Q{\Box}


\begin{cor}
Let $C = I_{X}$ be a group code defined over $F$. The extended code $\widetilde{C}^{\bot_{H}}$ is %%@
Hermitian self-dual if and only if $C$ is a split group code for some splitting $(Z = \{0 \}, X_{0}, %%@
X_{1})$ of $G$ by $-q$.
\label{Cor exte2}
\end{cor}
{\it Proof}. 

\noindent $(\Longleftarrow)$ This follows directly from the previous theorem.


\noindent $(\Longrightarrow)$ Since $\widetilde{C}^{\bot_{H}}$ is Hermitian self-dual, the dimension of %%@
$C$ is $\frac{n+1}{2}$ and so $C$ cannot be Hermitian self-orthogonal. This fact combined with the %%@
assumption that $\widetilde{C}^{\bot_{H}}$ is Hermitian self-dual implies that $0 \notin X$.
Let $C_{e} = I_{X \cup \{ 0 \} }$. This subcode $C_{e}$ is Hermitian self-orthogonal and has dimension %%@
$\frac{n-1}{2}$. By Theorem \ref{Thm HermSplit}, $C_{e}$ is a subcode of a split group code which is split %%@
by $-q$, that is, $C_{e} = C_{0}^{Z}$ for some splitting $(Z, X_{0}, X_{1})$ of $G$ by $-q$. Since dim %%@
$C_{e} = \frac{n-1}{2}$ and dim $C_{0}^{Z} = \frac{n- |Z|}{2}$, it follows that $Z = \{0 \}$. Hence $X = %%@
X_{0}$ and $C$ is a split group code of $G$ which is split by $-q$.
\hfill \Q{\Box}


\subsection{Existence of Hermitian Self-dual Extended Ideal Codes}

In view of Theorem \ref{Thm HermSplit} and Corollary \ref{Cor exte2}, it is natural to ask under what %%@
conditions we obtain splittings over $Z = \{ 0 \}$ of an abelian group $G$ by $-q$. Such conditions would %%@
guarantee existence of Hermitian self-orthogonal codes and Hermitian self-dual extended codes in $\FG$. We %%@
remark that the results in this section are generalizations of results on extended duadic codes in %%@
\cite{DMS}.

Define $ord_{r}(q)$ to be the smallest positive integer $t$ such that $q^{t} \equiv 1 \;(\mbox{mod } r)$. %%@
If $l$ is a positive odd integer relatively prime to $q$ then $l$ is said to be split by $-q$ over %%@
$\FF_{q^{2}}$ if and only if the set $X = \{1, 2, \ldots, l \}$ has a partition $X = X_{0} \cup X_{1}$ %%@
such that $(-q)X_{0} = X_{1}$ and $(-q)X_{1} = X_{0}$, where the multiplication is read modulo $l$.

\begin{prop} {\rm (\cite{DMS}.)}
Let $l$ be a positive odd integer which is relatively prime to $q$. The integer $l$ has a splitting by %%@
$-q$ if and only if $ord_{r}(q) \not\equiv 2 \;(\mbox{mod } 4)$ for every prime $r$ dividing $l$.
\label{duadic}
\end{prop}


\begin{thm}
Let $G$ be an abelian group of order $n$. The group $G$ has a splitting over $Z = \{ 0 \}$ given by $-q$ %%@
if and only if $ord_{r}(q) \not\equiv 2 \;(\mbox{mod } 4)$ for every prime $r$ dividing $n$.
\label{spli}
\end{thm}
{\it Proof}.
The abelian group $G$ is isomorphic to a unique product of cyclic groups of the form
\begin{equation*}
\ZZ_{m_{1}} \times \ZZ_{m_{2}} \times \ldots \times \ZZ_{m_{s}},
\end{equation*}
where $m_{i}$ divides $m_{i+1}$ for $i = 1, 2, \ldots, s-1$, and $m_{s} = m$ where $m$ denotes the %%@
exponent of $G$.

If each summand $\ZZ_{m_{i}}$ has a splitting over $Z = \{ 0 \}$ given by $-q$ then $G$ also has a %%@
splitting over $Z = \{ 0 \}$ given by $-q$. Indeed if $(\{ 0 \}, X_{0}^{(i)}, X_{1}^{(i)})$ is a splitting %%@
by $-q$ of $\ZZ_{m_{i}}$ for each $i = 1, 2, \ldots, s$ then letting
\begin{eqnarray*}
X_{t} & = & \hspace{.1in} X_{t}^{(1)} \times \ZZ_{m_{2}} \times \ZZ_{m_{3}} \times \cdots \times %%@
\ZZ_{m_{s}}\\
	  &   & \cup \; \{ 0 \} \times X_{t}^{(2)} \times \ZZ_{m_{3}} \times \cdots \times \ZZ_{m_{s}}\\
	  &   & \cup \; \{ 0 \} \times \{ 0 \} \times X_{t}^{(3)} \times \ZZ_{m_{4}} \times \cdots \times %%@
\ZZ_{m_{s}} \\
	  &   & \hspace{1in} \vdots \\
	  &   & \cup \; \{ 0 \} \times \{ 0 \} \times \{ 0 \} \times \cdots \times \{ 0 \} \times X_{t}^{(s)}
\end{eqnarray*}
for $t = 0, 1$, $(\{ 0 \}, X_{0}, X_{1})$ gives a splitting of $G$ by $-q$.
Conversely, suppose that $G$ has a splitting over $Z = \{ 0 \}$ given by $-q$. Let $\ZZ_{i} = \{ 0 \} %%@
\times \{ 0 \} \times \ldots \times \ZZ_{m_{i}} \times \{ 0 \} \times \ldots \{ 0 \}$ be the subgroup of %%@
$G$ isomorphic to $\ZZ_{m_{i}}$. If $(Z = \{0 \}, X_{0}, X_{1})$ gives a splitting for $G$ by $-q$, then %%@
$(Z = \{0 \}, \ZZ_{i} \cap X_{0}, \ZZ_{i} \cap X_{1})$ gives a splitting for $\ZZ_{i}$ given by $-q$.
Hence using Proposition \ref{duadic},
$G$ has a splitting over $Z = \{ 0 \}$ given by $-q$
if and only if each summand $\ZZ_{m_{i}}$ has a splitting over $Z = \{ 0 \}$ given by $-q$
if and only if $m_{i}$ is split by $-q$ for all $i = 1, 2, \ldots , m$
if and only if  $ord_{r}(q) \not\equiv 2 \;(\mbox{mod } 4)$ for every prime $r$ dividing $m_{i}$ for all %%@
$i = 1, 2, \ldots , s$
if and only if  $ord_{r}(q) \not\equiv 2 \;(\mbox{mod } 4)$ for every prime $r$ dividing $m$.
But the primes dividing $m$ are precisely the primes dividing $n$.
Thus $G$ has a splitting over $Z = \{ 0 \}$ given by $-q$ if and only if $ord_{r}(q) \not\equiv 2 %%@
\;(\mbox{mod } 4)$ for every prime $r$ dividing $n$.
\hfill \Q{\Box}

\vskip .1in

\noindent \emph{Example}:  Let $G = \ZZ_{3} \times \ZZ_{9}$ and $\FF_{4^{2}} = \FF_{16}$. Note that %%@
$ord_{3}(4) = 1$ and by Theorem \ref{spli} the abelian group $G$ has a partition which is split by $-4$. %%@
The cyclic groups $\ZZ_{3}$ and $\ZZ_{9}$ have splittings by the multiplier $\mu_{-4}$ given by $(\{0\}, %%@
A_{1}, A_{2})$ and $(\{0\}, B_{1} \cup B_{3}, B_{2} \cup B_{6})$, respectively, where $A_{i}$ is the %%@
$16$-cyclotomic coset modulo $3$ containing $i$ and $B_{j}$ is the $16$-cyclotomic coset modulo $9$ which %%@
contains $j$. Define $C_{(i,j)}$ as the orbit of $\tau_{16}$ in $G$ containing $(i,j)$. Letting $X_{0} = %%@
C_{(1,0)} \cup C_{(1,1)}  \cup C_{(1,2)} \cup C_{(1,3)}  \cup C_{(1,6)}  \cup C_{(0,1)}  \cup C_{(0,3)}$ %%@
and $X_{1} = C_{(2,0)} \cup C_{(2,1)}  \cup C_{(2,2)} \cup C_{(2,3)}  \cup C_{(2,6)}  \cup C_{(0,2)}  \cup %%@
C_{(0,6)}$, the set $(\{ (0,0) \}, X_{0}, X_{1} )$ gives a splitting of $G$ by $-4$. Notice that this %%@
partition can be obtained from the splittings of $\ZZ_{3}$ and $\ZZ_{9}$ as described in the proof.
\hfill \Q{\Box}

\vskip .1in

\noindent We remark that $ord_{r}(q) \not\equiv 2 \;(\mbox{mod } 4)$ means that either $ord_{r}(q)$ is odd %%@
or $ord_{r}(q)$ is doubly even. It can easily be verified that $ord_{r}(q)$ is doubly even if and only if %%@
$ord_{r}(q^{2})$ is even. Thus Theorem \ref{spli} can be restated as:

\begin{thm}
 Let $G$ be an abelian group of order $n$. The group $G$ has a splitting over $Z = \{ 0 \}$ given by $-q$ %%@
if and only if  for every prime $r$ dividing $n$,  either $ord_{r}(q)$ is odd or $ord_{r}(q^{2})$ is even.
\label{Thm spli2}
\end{thm}


\noindent Thus we get the following condition for the existence of extended ideal codes of $\FG$ which are %%@
Hermitian self-dual.


\begin{thm}
Let $G$ be an abelian group of order $n$.
An ideal code of $\FG$ whose extension is Hermitian self-dual exists if and only if for every prime $r$ %%@
dividing $n$,  either $ord_{r}(q)$ is odd or $ord_{r}(q^{2})$ is even.
\label{Thm concon2}
\end{thm}
{\it Proof}. This is a direct consequence of Corollary \ref{Cor exte2} and Theorem \ref{Thm spli2}.
\hfill \Q{\Box}

\vskip .1in

\noindent We note that the same result as the preceding theorem was obtained by Mart\'{i}nez-P\'{e}rez and %%@
Willems in \cite{conchita} for ideal codes in a group algebra over any finite group. 




\section{Counting Hermitian self-dual extended abelian group codes}
Theorem \ref{Thm concon2} raises the question of counting the number of non-isomorphic
abelian groups of order $\le x$ for which an ideal code of $F[G^*]$ whose
extension is Hermitian self-dual exists. An estimate for this quantity
is provided by Theorem \ref{twee}.

Let $q=p_1^t$ be a prime power and let ${\cal P}_q$ be the set of primes 
$r\ne p_{1}$ for which ord$_{r}(q)$ is odd or ord$_{r}(q^2)$ is even. 
Let ${\cal P}_q(x)$
be the associated counting function. The primes $r$ not counted, that
is the primes $r$ such that ord$_r(q)\equiv 2({\rm mod~}4)$  or the
prime $r=p_1$ can 
be shown, see \cite{DMS}, to have a natural density $\delta(q)$ that
is given by the following formula (with $\lambda$ the exponent of 2 in the
factorisation of $t$):
\begin{equation*}
\delta(q)=\delta(p_1^{t})=\left\{ 
\begin{array}{ll}
7/24                & \mbox{ if }  p_1=2 \mbox{ and }  \lambda=0 ; \\
1/3                 & \mbox{ if } p_1=2 \mbox{ and } \lambda=1 ; \\
2^{-\lambda -1}/3   & \mbox{ if } p_1=2 \mbox{ and } \lambda\ge 2 ;\\
2^{-\lambda}/3      & \mbox{ if } p_1\ne 2.
\end{array}
\right.
\end{equation*}
It can be proved, see \cite[Lemma A.3]{DMS}, that 
\begin{equation}
\label{estimate}
{\cal P}_q(x)=(1-\delta(q)){\rm Li}(x)+O_{q}\left(\frac{x(\log \log x)^4}{\log^3 x}\right),
\end{equation}
where the subscript $q$ indicates that the implied constant may depend on $q$ and
$Li(x)=\int_2^x dt/\log t$ denotes the logarithmic integral.

Let ${\cal G}_q$ be the subsemigroup of the natural numbers generated by the
primes in ${\cal P}_q$. Let $HSD(x)$ count the number of non-isomorphic abelian
groups of order $n$ with $(n,q)=1$ and $n\le x$ for
which an ideal code of $F[G^*]$ whose
extension is Hermitian self-dual exists. Then by Theorem \ref{Thm concon2} we have that
$$HSD(x)=\sum_{n\le x,~n\in {\cal G}_q}a(n),$$
where $a(n)$ denotes the number of non-isomorphic abelian groups 
having $n$ elements. 

Thus we are naturally led to study the behaviour of $a(n)$
on subsemigroups $\cal G$ of the natural numbers. For our purposes it is enough 
to restrict to subsemigroups $\cal G$ that are
generated by a set $\cal P$ of primes satisfying
\begin{equation}
\label{hallo}
{\cal P}(x)=\tau {\rm Li}(x)+E_{\cal P}(x),
\end{equation}
where $0<\tau<1$ and the error term $E_{\cal P}(x)$ is small enough.

Although the literature on $a(n)$ is quite extensive, the latter problem does not seem
to have been studied before. 
Before delving into it, we recall some relevant
facts on the behaviour of $a(n)$.

\subsection{Counting non-isomorphic abelian groups}
It is easy to see that $a(n)$ is a multiplicative function with the property
that $a(p^k)=P(k)$ for every prime $p$ and every integer $k\geq 1,$ where $P(k)$
denotes the number of unrestricted partitions of $k.$ Thus $a(p^k)$ does not
depend on $p$ but only on $k,$ so that $a(n)$ is a ``prime independent''
multiplicative function.

An analytic approach to $a(n)$ is based on the fact that the Dirichlet series
associated with this function may be written as products of the Riemann zeta
function, which is defined for $\Re(s)>1$ as 
$\zeta (s)= \sum_{n=1}^{\infty} n^{-s}= \prod_p {(1-p^{-s})}^{-1}$ and otherwise
by analytic continuation.
Using the well-known identity
$$\sum_{k=0}^{\infty} P(k)x^k= \prod_{m=1}^{\infty} {1\over {{1-x}^m}},~|x|
<1,$$
one finds that, for $\Re(s)>1,$
$$\sum_{k=0}^{\infty}{{a(p^k)}\over {p^{ks}}}=\sum_{k=0}^{\infty} {{P(k)}\over
{p^{ks}}}= \prod_{m=1}^{\infty} {1\over {1-{1\over {p^{ms}}}}},$$
and thus, using the multiplicativity of $a(n),$
$$\sum_{n=1}^{\infty} {{a(n)}\over {n^s}}= \prod_p \sum_{k=0}^{\infty}
{{a(p^k)}\over {p^{ks}}}= \prod_p \prod_{m=1}^{\infty} {1\over
{1-{1\over{p^{ms}}}}}= \prod_{m=1}^{\infty} \zeta (ms).$$
Using the standard results from tauberian theory, one obtains
$$\sum_{n\leq x} a(n) \sim x \prod_{m=2}^{\infty} \zeta (m), ~ x\to \infty,$$
from this.
By much more refined methods, it can be shown that
$$\sum_{n\leq x} a(n) = \sum_{m=1}^3 c_m x^{1/m}+E(x),~c_m= \prod_{\substack{k=1
\\ k\neq m}}^{\infty}
\zeta \left({k\over m}\right),$$
where the estimates for the error term $E(x)$ have a long history
of improvements, with the best result to date being due
to Robert and Sargos \cite{RS}, who  proved that $|E(x)|\ll x^{{1/4}+\epsilon}$.
Furthermore one has, see \cite[p. 274]{F}, $c_1=2.2948565916\cdots$, 
$c_2=-14.6475663016\cdots$ and $c_3=118.6924619727\cdots$.

Thus on average $a(n)$ is constant (namely 
about $2.29$). Individual values, however, might get large.
In this direction Kr\"atzel \cite{K} proved that
\begin{equation}
\label{kra}
\lim_{n\rightarrow \infty}\sup \log (a(n)){\log \log n\over \log n}={\log 5\over 4},
\end{equation}
which implies that $a(n)\ll n^{\epsilon}$ for every $\epsilon>0$.

Ivi\'c \cite{I1} has pointed out that $C(x)$, the number of {\it distinct} values assumed
by $a(n)$ for $n\le x$, satisfies the bound
\begin{equation}
\label{cx}
C(x)\le \exp((1+o(1))2\pi\sqrt{\log x/3\log \log x}).
\end{equation}
The reason for this (see \cite[pp.  130-131]{I1}) is that there are 
$$\exp((1+o(1))2\pi\sqrt{\log x/3\log \log x})$$
integers $n\le x$ of the form
\begin{equation}
\label{form}
n=2^{a_2}3^{a_3}\cdots p^{a_p},~a_2\ge a_3\ge \cdots \ge a_p\ge 1,
\end{equation}
which is a classical result of Hardy and Ramanujan \cite[pp. 245-261]{R}.
Suppose that $a(n)$ is counted by $C(x)$, and let
$$n=p_{1,1}^{b_1}\cdots p_{1,k}^{b_k},~b_1\ge b_2\ge \cdots \ge b_k\ge 1,$$
be the canonical decomposition of $n$. Then if $m=2^{b_1}3^{b_2}\cdots p_k^{b_k}$, we
have $m\le n$ and $a(m)=P(b_1)\cdots P(b_k)=a(n)$. Therefore $C(x)$ does not
exceed the number of $n\le x$ having the form (\ref{form}) and hence
inequality (\ref{cx}) holds.

Note that if $f$ is {\it any} prime independent function, then the number of distinct
 values assumed by it for $n\le x$ satisfies the same
 upperbound as in (\ref{cx}).


\subsection{Summing $a(n)$ over $\cal G$}
Let $\chi_{\cal G}$ be the characteristic
function of $\cal G$, i.e.,
$$\chi_{\cal G}(n)=\begin{cases}
1 & {\rm if~}n{\rm ~is~in~}{\cal G};\cr 
0 & {\rm otherwise}.
\end{cases}
$$ 
We consider
$$\sum_{n\le x,~n\in {\cal G}}a(n)=\sum_{n\le x}\chi_{\cal G}(n)a(n).$$
Note that $\chi_{\cal G}(n)a(n)$ is multiplicative in $n$.
\begin{thm}
\label{blup}
If {\rm (\ref{hallo})} is 
satisfied with $E_{\cal P}(x)=O(x\log^{-1-\gamma}x)$ and $0<\gamma<1$, then
$$
\sum_{n\le x,~n\in {\cal G}}a(n)=xb_{0}\log^{\tau-1}x+O_{\cal G}(x\log^{\tau-1-\gamma/2}x).
$$
If {\rm (\ref{hallo})} is 
satisfied with $E_{\cal P}(x)=O(x\log^{-2-\gamma}x)$ and $\gamma>0$.
Then
\begin{equation}
\sum_{n\le x,~n\in {\cal G}}a(n)=
x\sum_{0\le \nu<\gamma}b_{\nu}\log^{\tau-1-\nu}x+O_{\cal G}(x\log^{\tau-1-\gamma+\epsilon}x),
\end{equation}
where $b_0,b_1,\ldots$ are constants possibly depending on $\cal G$ and
$$b_0={1\over \Gamma(\tau)}\lim_{s\downarrow 1}(s-1)^{\tau}\sum_{n\in {\cal G}}{a(n)\over n^{s}}>0.$$
\end{thm}
The proof uses the following lemma, which except for the formula for $b_0$ is
taken from \cite{MC}. The formula for $b_0$ is well-known.
\begin{lem} {\rm \cite{MC}}.
\label{een}
\label{moca}
Let $f:\mathbb N_{\ge 0}\rightarrow \mathbb R_{\ge 0}$ be a multiplicative
function satisfying
\begin{equation}
\label{groei}
0\le f(p^r)\le c_1c_2^r,~c_1\ge 1,~1\le c_2<2,
\end{equation}
and
\begin{equation}
\label{groei2}
\sum_{p\le x}f(p)=\tau {\rm Li}(x)+O({x\log^{-2-\gamma}x}),
\end{equation}
where $\tau>0$ and $\gamma>0$ are fixed, then, for $\epsilon>0$,
$$
\sum_{n\le x}f(n)=x\sum_{0\le \nu<\gamma}b_{\nu}\log^{\tau-1-\nu}x+O(x\log^{\tau-1-\gamma+\epsilon}x),
$$
where $b_0={1\over \Gamma(\tau)}\lim_{s\downarrow 1}(s-1)^{\tau}\sum_{n=1}^{\infty}{f(n)n^{-s}}$.
\end{lem}

\noindent {\it Proof of Theorem} \ref{blup}. The first assertion has been proved
by Odoni \cite{O} using a tauberian remainder theorem due to Subhankulov.
\hfill \Q{\Box}

In order to prove the second assertion we apply Lemma \ref{een} 
with $f(n)=a(n)\chi_{\cal G}(n)$. The fact that 
condition (\ref{groei}) is satisfied
follows from the classical result of Hardy and Ramanujan (see \cite[p. 240]{R}), that
$a(p^r)=P(r)=(1+o(1))(4\sqrt{3}r)^{-1}e^{\pi\sqrt{2r/3}}$ as $r$ tends to infinity.
However, the much more easily proved upperbound $P(r)\le 5^{r/4}$, see \cite{K}, is already 
sufficient in order
to show that (\ref{groei}) is satisfied.
The assumption on $E_{\cal P}(x)$ ensures that condition (\ref{groei2}) is satisfied.
On invoking Lemma \ref{moca} the proof is then completed.
   
For our problem at hand we the find the following estimate:
\begin{thm}
\label{twee}
Let $HSD(x)$ count the number of non-isomorphic abelian
groups of order $n$ with $(n,q)=1$ and $n\le x$ for
which an ideal code of $F[G^*]$ whose
extension is Hermitian self-dual exists. Then
$$HSD(x)=b_0{x\over \log^{\delta(q)}x}+O_{\epsilon,q}\left({x\over \log^{\delta(q)+1-\epsilon}x}\right),$$
where $$b_0={1\over \Gamma(1-\delta(q))}\lim_{s\downarrow 1}(s-1)^{1-\delta(q)}
\sum_{n\in {\cal G}}{a(n)\over n^{s}}.$$
\end{thm}

\subsection{The connection with free arithmetical semigroups}
A much weaker form of Theorem \ref{twee} is obtained as a straightforward consequence of
Bredikhin's Theorem, which is a basic result in the theory of {\it free arithmetical
semigroups}.

Let $G$ be a commutative semigroup with identity element 1, relative to a multplication operation
denoted by juxtaposition. Suppose that $G$ has a finite or countably infinite subset $P$ of
generators and that $G$ is {\it free}. This means that every element $n$ in $G$ has a unique
factorisation of the form $n=\omega_1^{a_1}\cdot \omega_2^{a_2}\cdots \omega_r^{a_r}$, where
the $\omega_r$ are distinct elements of $P$, the $a_i$ are possible integers, and uniqueness is
 up to order of factors. A free semigroup will be called a free arithmetical semigroup if in addition
 there exists a homomorphism of $G$ into some multiplicative semigroup ${\overline G}$ consisting
 of real numbers such that for every $x>0$, $G$ contains only finitely many elements $n$ with
 $|n|\le x$, where $|n|$ denotes the image (or norm) of the element $n$ of $G$ under the
 homomorphism $|.|$. (In the older literature the generators of ${\overline G}$ are called
 Beurling's generalized primes.) Bredikhin's theorem, for
 a proof see e.g. \cite[pp. 92-99]{Po}, then reads as follows:
\begin{thm} (Bredikhin.)
If $G$ is a free arithmetical semigroup such that
\begin{equation}
\label{weereen}
\sum_{|\omega|\le x,~\omega\in G}1=\tau{x\over \log x}+O\left({x\over \log^{1+\gamma}x}\right),
\end{equation}
where $\tau>0$ and $\gamma>0$ are fixed, then
$$
\sum_{|n|\le x\atop n\in G}1=C_Gx\log^{\tau-1}x+O(x\log^{\tau-1}x(\log \log x)^{-\gamma_1}),$$
where $\gamma_1=\min (1,\gamma)$ and
$C_G={\Gamma(\tau)}^{-1}\lim_{s\downarrow 1}(s-1)^{\tau}\sum_{n\in G}|n|^{-s}$.
\end{thm}

Now consider the free 
arithmetical semigroup $G$ of all non-isomorphic finite abelian groups with as composition the
usual direct product operation and as norm function $|A|={\rm card}(A)$. By the fundamental
theorem on finite abelian groups, $G$ is a free arithmetical semigroup having
$C(p),~C(p^2),~C(p^3),\cdots$ as generators, where $p$ runs over all the primes and $C(n)$ is
the cyclic group of order $n$. 
Since the number of cyclic groups of prime power order whose norm is not prime having
norm $\le x$ is $O(\sqrt{x}\log x)$, by the prime number theorem
in the form $\pi(x)=x/\log x + O(x/\log^2 x)$, (\ref{weereen}) is satisfied with 
$\tau=1$ and $\gamma=1$. It then follows from Bredikhin's theorem that
$$\sum_{|n|\le x\atop n\in G}1=\sum_{n\le x}a(n)=x\prod_{m=2}^{\infty}\zeta(m)+
O\left({x\over \log\log x}\right),$$
where we have used the observations that $\sum_{n\in G}|n|^{-s}=
\sum_n a(n)n^{-s}=\prod_{m=1}^{\infty}\zeta(ms)$
and $\lim_{s\downarrow 1}(s-1)\zeta(s)=1$.

Now let $G_q$ be the free arithmetical semigroup generated by all cyclic groups of the
form $C(p),C(p^2),C(p^3),\ldots$, with ord$_p(q)$ is odd or ord$_p(q^2)$ is even. Then
similarly using Bredikhin's theorem we obtain the result in Theorem \ref{twee} with the much weaker
error term $O_q(x\log^{-\delta(q)}x (\log \log x)^{-1})$.

\subsection{The maximal order of $a(n)$ on $\cal G$}
In this section we indicate what Kr\"atzel's result (\ref{kra}) looks like when one
considers the maximal order of $a(n)$ on the subsemigroup $\cal G$.
\begin{thm}
\label{drie}
Let $A=A(n)$ be the smallest integer such that $$\sum_{p\in {\cal P},~p\le A}\log p\ge 
{(\log n)}/4.$$
Then as $n$ tends to infinity and runs through the elements of $\cal G$, the estimate
$$\log a(n) \le {\cal P}(A)\log 5+O({\cal P}(A^{\theta})\log A),$$
holds with $\theta=\log(121)/\log(125)<0.994$, and there are infinitely many integers
$n$, $n\in G$, for which one has $\log a(n) ={\cal P}(A)\log 5$.
\end{thm}
{\it Proof}. Completely similar to that of the
(only) theorem in Schwarz and Wirsing \cite{SW}, who proved
this result in case $\cal P$ is the full set of primes. In their proof one merely
intersects every range of primes that occurs with $\cal P$.
\hfill \Q{\Box}

\vskip .1in
\noindent {\tt Remark}. The implicit constant in the order term can taken to be
$${2\pi^2\over 3\log 5\cdot \log 2}=5.898\cdots$$
\begin{thm}
\label{klaarhoor}
If ${\cal P}(x)\sim \tau x/\log x$ as $x$ tends to infinity, then
$$\lim_{n\rightarrow \infty}\sup_{n\in {\cal G}} 
\log (a(n)){\log \log n\over \log n}={\log 5\over 4}.$$
\end{thm}
{\it Proof}. By a standard argument in elementary
number theory it follows that if ${\cal P}(x)\sim \tau x/\log x$, 
then $\sum_{p\in {\cal P},~p\le x}\log p\sim \tau x$, $A(n)\sim (\log n)/(4\tau)$ and
${\cal P}(A)\sim \log n/(4\log \log n)$. On invoking Theorem \ref{drie} the
result then follows.
\hfill \Q{\Box}

\vskip .1in
\noindent {\tt Remark}. Let $p_1,p_2,\cdots$ denote the consecutive primes in $\cal P$.
Let $n_r=\prod_{i=1}^r p_i^4$. Suppose that ${\cal P}(x)\sim \tau x/\log x$ as $x$ tends
to infinity. We leave it as an exercise to the reader to show that
$$\lim_{r\rightarrow \infty}\log(a(n_r)){\log \log n_r\over \log n_r}={\log 5\over 4}.$$

\noindent {\tt Remark}. It is rather surprising that in Theorem \ref{klaarhoor} the
estimate does not depend on $\tau$. A similar situation arises if one compares the
maximal order of $\log d(n)$ with that of $\log r(n)$, where $d(n)$ denotes the number
of divisors of $n$ and $r(n)$ the number of way $n$ can be written as a sum of two
squares. Jacobi proved that $r(n)=4\{d_1(n)-d_3(n)\}$, where $d_1(n)$ and $d_3(n)$ denote
the number of the divisor of $n$ of the form $4k+1$ and $4k+3$, respectively. Thus
$r(n)$ counts (crudely) the divisors of $n$ made up of prime $\equiv 1({\rm mod~}4)$.
These primes have density $1/2$ amongst all primes, but nevertheless the maximal
orders of $\log d(n)$ and $\log r(n)$ are the same. Namely, we have
$$\lim_{n\rightarrow \infty}\sup \log (d(n)){\log \log n\over \log n}={\log 2},~
\lim_{n\rightarrow \infty}\sup \log (r(n)){\log \log n\over \log n}={\log 2}.$$
For further details see e.g. Nicolas \cite{Nic}. The maximal order for $\log d(n)$
was first determined by S. Wigert in 1907. Hardy and Wright \cite[Theorem 338]{HW} 
erroneously give $(\log 2)/2$ instead of $\log 2$ in the result for $\log r(n)$.



\subsection{Counting distinct values assumed by $a(n)$ on $\cal G$}
Let $C_{\cal G}(x)$ denote the number of distinct values assumed by $a(n)$
with $n\in {\cal G}$ and $n\le x$.
\begin{thm}
Let $p_0$ be the smallest prime in $\cal P$.
Suppose that there are positive constants $c_3$ and $c_4$ such that, for $x\ge p_0$,
$$c_3x< \sum_{p\in {\cal P},~p\le x}\log p<c_4x,$$
then 
$$\log C_{\cal G}(x)\sim \log C(x)\sim (1+o(1))2\pi\sqrt{\log x/3\log \log x},$$
as $x$ tends to infinity.
\end{thm}
{\it Proof}. Very similar to that given in \cite[pp. 245-261]{R}. Instead of
defining $l_n$ to be the product of the first $n$ consecutive primes, we define
it to be the product of the first $n$ consecutive primes in $\cal P$. Then instead
of (3.23) we find
$\phi(s)>c_1\int_{p_0}^{\infty}e^{-c_1sx}dx/\log x+O(1)$ and instead
of (3.24) we find $\phi(s)<c_2\int_{p_0}^{\infty}e^{-c_2sx}dx/\log x+O(1)$.
This, through Lemma 3.4, then leads to the same asymptotic for $\phi(s)$
as in the paper of Hardy and Ramanujan. This then results in the same
asymptotic for $C_{\cal G}(x)$ as that for $C(x)$.
\hfill \Q{\Box}


\vskip .3in
\noindent {\bf Acknowledgements}. The first author gratefully acknowledges financial support from the %%@
University of the Philippines and from the Philippine Council for Advanced Science and Technology Research %%@
and Development through the Department of Science and Technology.

The second author would like to thank Alexander Ivi\'c for pointing out reference \cite{RS} to him.

{\small
\begin{thebibliography}{DGS}


\bibitem{intro in handbook} R.A.~Brualdi, W.C.~Huffman, V.S.~Pless, \emph{An Introduction to Algebraic %%@
Codes}, in \emph{Handbook of Coding Theory}, V.S.~Pless \& W.C.~Huffman (Editors), Elsevier Science, %%@
Amsterdam (1998), pp. 3-139.

\bibitem{DMS} L.~Dicuangco, P.~Moree, P.~Sol\'e, \emph{The Lengths of Hermitian Self-dual Extended Duadic %%@
Codes}. Preprint (2005), arXiv:math.CO/0511295, submitted.

\bibitem{cunsheng} C.~Ding, D.R.~Kohel, S.~Ling, \emph{Split Group Codes}. IEEE Transactions on %%@
Information Theory, Vol. {\bf IT-46} (2000), pp. 485-495.

\bibitem{F} S.R.~Finch, \emph{Mathematical constants}. 
Encyclopedia of Mathematics and its Applications {\bf 94}, Cambridge University Press, 
Cambridge, 2003.

\bibitem{HW}  G.H.~Hardy, E.M.~Wright, 
\emph{An introduction to the theory of numbers}. Fifth edition. The Clarendon Press, Oxford University Press, 
New York, 1979

\bibitem{fundamentals} W.C.~Huffman, V.~Pless, \emph{Fundamentals of Error-Correcting Codes}. Cambridge %%@
University Press, 2003.

\bibitem{hungerford} T.W.~Hungerford, \emph{Algebra}. Springer-Verlag, New York, 1974.

\bibitem{IR} K.~Ireland and M.~Rosen, \emph{A Classical Introduction to Modern Number
Theory (Second Edition)}. Graduate Texts in Mathematics {\bf 84},
Springer-Verlag, New York, 1990.

\bibitem{I1} A.~Ivi\'c, \emph{On the number of abelian groups of a given order and on certain related %%@
multiplicative functions}. J. Number Theory, {\bf 16}  (1983), pp. 119-137. 

\bibitem{K} E.~Kr\"atzel, \emph{Die maximale Ordnung der Anzahl der wesentlich verschiedenen 
abelschen Gruppen $n$-ter Ordnung}. Quart. J. Math. Oxford Ser, {\bf 21} (1970), pp. 273-275. 
 
\bibitem{leon et al} J.S.~Leon, J.M.~Masley, V.~Pless, \emph{Duadic Codes}. IEEE Transactions on %%@
Information Theory, Vol. {\bf IT-30}, No. 5 (1984), pp. 709-714.

\bibitem{SD over GF4} F.J.~MacWilliams, A.M.~Odlyzko, N.J.A.~Sloane, H.N.~Ward, \emph{Self-Dual Codes %%@
over GF(4)}. Journal of Combinatorial Theory, Series {\bf A 25} (1978), pp. 288-318.

\bibitem{theory} F.J.~MacWilliams, N.J.A.~Sloane, \emph{The Theory of Error-Correcting Codes}. North %%@
Holland Publishing Company, Amsterdam (1983).

\bibitem{conchita} C.~Mart\'{i}nez-P\'{e}rez, W.~Willems, \emph{Self-Dual Extended Cyclic Codes}. Preprint %%@
(2004), http://www.math.uni-magdeburg.de/preprints/shadows/04-30report.html.

\bibitem{moree} P.~Moree, \emph{On the divisors of $a^{k} + b^{k}$}. Acta Arithmetica {\bf 80}, No. 3 %%@
(1997), pp. 197-212.

\bibitem{Moree2} P.~Moree, \emph{On primes $p$ for which $d$ divides ord$_p(g)$}.
Funct. Approx. Comment. Math. {\bf 33} (2005), pp. 85-95.

\bibitem{MC} P.~Moree and J.~Cazaran, \emph{On a claim of Ramanujan in his first letter to Hardy}. %%@
Exposition. Math., {\bf 17} (1999), pp. 289-311.

\bibitem{Nic} J.-L.~Nicolas, \emph{On highly composite numbers}, in \emph{Ramanujan revisited} 
(Urbana-Champaign, Ill., 1987), Academic Press, Boston, MA, 1988, pp. 215--244.

\bibitem{O}  R.W.K.~Odoni, \emph{A problem of Rankin on sums of powers of cusp-form coefficients}. 
J. London Math. Soc. (2), {\bf  44}  (1991), pp. 203-217. 

\bibitem{qcodes} V.~Pless, \emph{Q-Codes}. Journal of Combinatorial Theory, Series {\bf A 43} (1986), pp. %%@
258-276.

\bibitem{onweights} V.~Pless, J.M.~Masley, J.S.~Leon, \emph{On Weights in Duadic Codes}. Journal of %%@
Combinatorial Theory, Series {\bf A 44} (1987).

\bibitem{Po} A.G.~Postnikov, \emph{Introduction to analytic number theory}. Translations of Mathematical %%@
Monographs {\bf 68}, AMS, Providence, RI, 1988. 

\bibitem{R} S.~Ramanujan, \emph{Collected papers}. Chelsea, New York, 1962.

\bibitem{rushanan} J.J.~Rushanan, \emph{Duadic Codes and Difference Sets}. Journal of Combinatorial %%@
Theory, Series {\bf A 57} (1991), pp. 254-261.

\bibitem{RS}  O.~Robert and P.~Sargos, \emph{Three-dimensional exponential sums with monomials}. J. Reine %%@
Angew. Math. {\bf 591} (2006), pp. 1-20.

\bibitem{SW} W.~Schwarz and E.~Wirsing, \emph{The maximal number of non-isomorphic abelian groups of order %%@
$n$}. Arch. Math. (Basel),  {\bf 24}  (1973), pp. 59-62.

\bibitem{smid} M.~Smid, \emph{Duadic Codes}. IEEE Transactions on Information Theory, Vol. {\bf IT-33} No. %%@
3 (1987), pp. 432-433.

\bibitem{smidthesis} M.~Smid, \emph{On Duadic Codes}. Master's Thesis, Eindhoven University of Technology, %%@
The Netherlands, 1986.

\bibitem{ward} H.N.~Ward, \emph{Quadratic Residue Codes and Divisibility}, in \emph{Handbook of Coding %%@
Theory}, V.S.~Pless \& W.C.~Huffman (Editors), Elsevier Science, Amsterdam (1998), pp. 827-870.

\bibitem{partitions} H.N.~Ward, L.~Zhu, \emph{Existence of Abelian Group Code Partitions}. Journal of %%@
Combinatorial Theory Series {\bf A 67} (1994), pp. 276-281.

\end{thebibliography}}

\end{document}






math0604013.extracted_bib
\begin{thebibliography}{DGS}


\bibitem{intro in handbook} R.A.~Brualdi, W.C.~Huffman, V.S.~Pless, \emph{An Introduction to Algebraic %%@
Codes}, in \emph{Handbook of Coding Theory}, V.S.~Pless \& W.C.~Huffman (Editors), Elsevier Science, %%@
Amsterdam (1998), pp. 3-139.

\bibitem{DMS} L.~Dicuangco, P.~Moree, P.~Sol\'e, \emph{The Lengths of Hermitian Self-dual Extended Duadic %%@
Codes}. Preprint (2005), arXiv:math.CO/0511295, submitted.

\bibitem{cunsheng} C.~Ding, D.R.~Kohel, S.~Ling, \emph{Split Group Codes}. IEEE Transactions on %%@
Information Theory, Vol. {\bf IT-46} (2000), pp. 485-495.

\bibitem{F} S.R.~Finch, \emph{Mathematical constants}. 
Encyclopedia of Mathematics and its Applications {\bf 94}, Cambridge University Press, 
Cambridge, 2003.

\bibitem{HW}  G.H.~Hardy, E.M.~Wright, 
\emph{An introduction to the theory of numbers}. Fifth edition. The Clarendon Press, Oxford University Press, 
New York, 1979

\bibitem{fundamentals} W.C.~Huffman, V.~Pless, \emph{Fundamentals of Error-Correcting Codes}. Cambridge %%@
University Press, 2003.

\bibitem{hungerford} T.W.~Hungerford, \emph{Algebra}. Springer-Verlag, New York, 1974.

\bibitem{IR} K.~Ireland and M.~Rosen, \emph{A Classical Introduction to Modern Number
Theory (Second Edition)}. Graduate Texts in Mathematics {\bf 84},
Springer-Verlag, New York, 1990.

\bibitem{I1} A.~Ivi\'c, \emph{On the number of abelian groups of a given order and on certain related %%@
multiplicative functions}. J. Number Theory, {\bf 16}  (1983), pp. 119-137. 

\bibitem{K} E.~Kr\"atzel, \emph{Die maximale Ordnung der Anzahl der wesentlich verschiedenen 
abelschen Gruppen $n$-ter Ordnung}. Quart. J. Math. Oxford Ser, {\bf 21} (1970), pp. 273-275. 
 
\bibitem{leon et al} J.S.~Leon, J.M.~Masley, V.~Pless, \emph{Duadic Codes}. IEEE Transactions on %%@
Information Theory, Vol. {\bf IT-30}, No. 5 (1984), pp. 709-714.

\bibitem{SD over GF4} F.J.~MacWilliams, A.M.~Odlyzko, N.J.A.~Sloane, H.N.~Ward, \emph{Self-Dual Codes %%@
over GF(4)}. Journal of Combinatorial Theory, Series {\bf A 25} (1978), pp. 288-318.

\bibitem{theory} F.J.~MacWilliams, N.J.A.~Sloane, \emph{The Theory of Error-Correcting Codes}. North %%@
Holland Publishing Company, Amsterdam (1983).

\bibitem{conchita} C.~Mart\'{i}nez-P\'{e}rez, W.~Willems, \emph{Self-Dual Extended Cyclic Codes}. Preprint %%@
(2004), http://www.math.uni-magdeburg.de/preprints/shadows/04-30report.html.

\bibitem{moree} P.~Moree, \emph{On the divisors of $a^{k} + b^{k}$}. Acta Arithmetica {\bf 80}, No. 3 %%@
(1997), pp. 197-212.

\bibitem{Moree2} P.~Moree, \emph{On primes $p$ for which $d$ divides ord$_p(g)$}.
Funct. Approx. Comment. Math. {\bf 33} (2005), pp. 85-95.

\bibitem{MC} P.~Moree and J.~Cazaran, \emph{On a claim of Ramanujan in his first letter to Hardy}. %%@
Exposition. Math., {\bf 17} (1999), pp. 289-311.

\bibitem{Nic} J.-L.~Nicolas, \emph{On highly composite numbers}, in \emph{Ramanujan revisited} 
(Urbana-Champaign, Ill., 1987), Academic Press, Boston, MA, 1988, pp. 215--244.

\bibitem{O}  R.W.K.~Odoni, \emph{A problem of Rankin on sums of powers of cusp-form coefficients}. 
J. London Math. Soc. (2), {\bf  44}  (1991), pp. 203-217. 

\bibitem{qcodes} V.~Pless, \emph{Q-Codes}. Journal of Combinatorial Theory, Series {\bf A 43} (1986), pp. %%@
258-276.

\bibitem{onweights} V.~Pless, J.M.~Masley, J.S.~Leon, \emph{On Weights in Duadic Codes}. Journal of %%@
Combinatorial Theory, Series {\bf A 44} (1987).

\bibitem{Po} A.G.~Postnikov, \emph{Introduction to analytic number theory}. Translations of Mathematical %%@
Monographs {\bf 68}, AMS, Providence, RI, 1988. 

\bibitem{R} S.~Ramanujan, \emph{Collected papers}. Chelsea, New York, 1962.

\bibitem{rushanan} J.J.~Rushanan, \emph{Duadic Codes and Difference Sets}. Journal of Combinatorial %%@
Theory, Series {\bf A 57} (1991), pp. 254-261.

\bibitem{RS}  O.~Robert and P.~Sargos, \emph{Three-dimensional exponential sums with monomials}. J. Reine %%@
Angew. Math. {\bf 591} (2006), pp. 1-20.

\bibitem{SW} W.~Schwarz and E.~Wirsing, \emph{The maximal number of non-isomorphic abelian groups of order %%@
$n$}. Arch. Math. (Basel),  {\bf 24}  (1973), pp. 59-62.

\bibitem{smid} M.~Smid, \emph{Duadic Codes}. IEEE Transactions on Information Theory, Vol. {\bf IT-33} No. %%@
3 (1987), pp. 432-433.

\bibitem{smidthesis} M.~Smid, \emph{On Duadic Codes}. Master's Thesis, Eindhoven University of Technology, %%@
The Netherlands, 1986.

\bibitem{ward} H.N.~Ward, \emph{Quadratic Residue Codes and Divisibility}, in \emph{Handbook of Coding %%@
Theory}, V.S.~Pless \& W.C.~Huffman (Editors), Elsevier Science, Amsterdam (1998), pp. 827-870.

\bibitem{partitions} H.N.~Ward, L.~Zhu, \emph{Existence of Abelian Group Code Partitions}. Journal of %%@
Combinatorial Theory Series {\bf A 67} (1994), pp. 276-281.

\end{thebibliography}