Dualities for codes over finite abelian groups (Q6605893)

The MacWilliams relations allow the weight enumerator of a linear code to be described by the weight enumerator of its dual code. The author restricts his research to abelian groups as alphabets because codes over non-abelian groups are known to lack MacWilliams relations. The orthogonality is not always symmetric, so this paper offers a very general setting regarding the coding theory. In this work the number of dualities, the double orthogonal condition, and the MacWilliams relations, in this context, are examined. This leads to the presentation of fundamental insights concerning the orthogonal code for symmetric and non-symmetric dualities. Descriptions for all possible dualities for any finite abelian group are established.\N\NFor a group \(G,\) the set of all characters of \(G\) is denoted by \(\widehat{G}.\) For example, given a finite abelian group \(G\), it's shown that the number of isomorphisms between \(G\) and \(\widehat{G}\) is the cardinality of \(\Aut(G),\) the group of automorphisms of the group \(G.\) Also, the number of symmetric dualities is the cardinality of the subset of \(\Aut(G)\) of automorphisms that are their own inverses.\N\NFor an additive group \(G\) with duality \(M\), and an additive code \(C\) of length \(n,\) the classic formula \(|C||C^M| = |G^n|\) is proven as well as MacWilliams relations for the symmetrized weight enumerator of the code. Various dualities for specific groups are investigated. Specific theorems, giving all possible dualities, and for some dualities, equivalences among the dualities, are provided.