  The "Nicolas-Serre code", $(a,b) \leftrightarrow t^{n}$, is a bijection
between $N\times N$ and those $t^{n}$, $n$ odd, in $Z/2[t]$. Suppose $A_{n}$,
$n$ odd, in $Z/2[t]$ are defined by: $A_{1}= A_{5}= 0$, $A_{3}= t$, $A_{7}=
t^{5}$, and $A_{n+8}= t^{8} A_{n} + t^{2} A_{n+2}$. A lemma, Proposition 4.3 of
[6], used to study the Hecke algebra attached to the space of mod $2$ level $1$
modular forms, gives information about the codes $(a,b)$ attached to the
monomials appearing in $A_{n}$. The unpublished highly technical proof has been
simplified by Gerbelli-Gauthier.
  Our Theorem 3.7 generalizes Proposition 4.3. The proof, in sections 1-3, is a
further simplification of Gerbelli-Gauthier's argument. We build up to the
theorem with variants involving the same recurrence, but having different sorts
of initial conditions.
  Section 4 treats the recurrence $A_{n+16}= t^{16} A_{n} + t^{4} A_{n+4} +
t^{2} A_{n+2}$. Theorem 4.1, the analog to Theorem 3.7 for this recurrence, is
used in [2] and [3] to analyze level 3 Hecke algebras.
  Finally we introduce a variant code, $(a,b) \leftrightarrow w^{n}$ which is a
bijection between $N\times N$ and those $w^{n}$, $n \equiv 1,3,7,9 \bmod{20}$,
in $Z/2[w]$. We then study the recurrence $A_{n+80}= w^{80} A_{n}+ w^{20}
A_{n+20}$, $n \equiv 1,3,7,9 \bmod{20}$, with appropriate initial conditions.
Lemma 5.5, derived from the results of sections 1-3, is the precise analog of
Proposition 4.3 for this code, this recurrence, and these initial conditions.
It is used in [4] and [5] to analyze level 5 Hecke algebras.
