Self-Dual Codes Over Chain Rings

Abstract

In this paper, we study self-dual codes over commutative Artinian chain rings. Let R be such a ring, x be a generator of the unique maximal ideal of R and $$a\in {\mathbb {N}}_0 $$ maximal such that $$x^a\ne 0$$. A code C over R of length t is an R-submodule of the free module $$R^t$$. Multiplying powers of x to C defines the finite chain of subcodes $$\begin{aligned} C \supseteq C^{(1)} := C x \supseteq C^{(2)} := C x^2 \supseteq \dots \supseteq C^{(a)} := Cx^a \supseteq \lbrace 0 \rbrace . \end{aligned}$$In this paper, we show that if C is a self-dual code in $$R^t$$, then $$C^{(a)}$$ is a (hermitian) self-dual code over the residue field $${\mathbb {F}}= R / \langle x \rangle $$ if and only if C a free R-module (thus isomorphic to $$R^{\frac{t}{2}}$$). In this case, all codes $$C^{(i)}$$ are self-dual codes in suitable bilinear or Hermitian spaces $$W_i$$ over $${\mathbb {F}}$$ and we describe a method to construct all lifts C of a given self-dual code $$C^{(a)}$$ over $${\mathbb {F}}$$ that are self-dual, free codes over R. We apply this technique to codes over finite fields of characteristic p admitting an automorphism whose order is a power of p. For illustration, we show that the well-known Pless Code $$P_{36}$$ is the only extremal, ternary code of length 36 with an automorphism of order 3, strengthening a result of Huffman, who showed the assertion for all prime orders $$\ge 5$$.