Automorphism Groups of Convolutional Codes
P. Delsarte, Ph. Piret · SIAM Journal on Applied Mathematics · 1978
Let K be the monomial group of degree n, over the field $F = GF( q )$, and let $K^\infty $ denote the group of mappings $x:\mathbb{Z} \to K:i \mapsto x^{( i )} $. For any sequence $v ( D ) = \sum {v_i D^i } $, with $v_i \in F^n $, and any x in $K^\infty $, the x-image of $v( D )$ is defined to be $v( D )x = \sum {v_i x^{( i )} D^i } $. Given a q-ary convolutional code C of length n, the set ${\operatorname{Aut}}( C )$ of all x satisfying $Cx = C$ is a subgroup of $K^\infty $, called the automorphism group of C. This can be viewed as the group of eternal walks in a certain finite directed graph. Let G be a shift-invariant subgroup of $K^\infty $. A convolutional code C is called a G-code whenever $G \leqq {\operatorname{Aut}}( C )$ holds. If the words of C form a completely reducible module over the group algebra $FG^{( 0 )} $, then C is the direct sum of irreducible G-codes generated by normal sequences, and these yield a minimal basis for C.