  In the last decade, there has been a continued effort to produce families of
strong linearizations of a matrix polynomial $P(\lambda)$, regular and
singular, with good properties. As a consequence of this research, families
such as the family of Fiedler pencils, the family of generalized Fiedler
pencils (GFP), the family of Fiedler pencils with repetition, and the family of
generalized Fiedler pencils with repetition (GFPR) were constructed. In
particular, one of the goals was to find in these families structured
linearizations of structured matrix polynomials. For example, if a matrix
polynomial $P(\lambda)$ is symmetric (Hermitian), it is convenient to use
linearizations of $P(\lambda)$ that are also symmetric (Hermitian). Both the
family of GFP and the family of GFPR contain block-symmetric linearizations of
$P(\lambda)$, which are symmetric (Hermitian) when $P(\lambda)$ is. Now the
objective is to determine which of those structured linearizations have the
best numerical properties. The main obstacle for this study is the fact that
these pencils are defined implicitly as products of so-called elementary
matrices. In this paper we consider the family of block-minimal bases pencils,
whose pencils are defined in terms of their block-structure, as a source of
canonical forms for block-symmetric pencils. More precisely, we present four
families of block-symmetric pencils which, under some generic nonsingularity
conditions are block minimal bases pencils and strong linearizations of a
matrix polynomial. We show that the block-symmetric GFP and GFPR, after some
row and column permutations, belong to the union of these four families. Hence,
these four families of pencils provide an alternative but explicit approach to
the block-symmetric Fiedler-like pencils existing in the literature.
