Dualities for finite abelian groups and applications to coding theory
Jay A. Wood
TL;DR
The paper develops a unified framework for dualities between a finite abelian group $A$ and its character group $\widehat{A}$ to define duals of additive codes over $A$, extending Delsarte's symmetric inner products to nonsymmetric ones. It analyzes how dual codes depend on the chosen duality via the automorphism group $\mathrm{Aut}(A)$, introduces a congruence relation among dualities, and derives MacWilliams identities for both the Hamming weight and the complete enumerator using Fourier analysis and Poisson summation. It further investigates structural questions, including when duals are invariant under all dualities and how a prime-by-prime decomposition $A=\bigoplus_p A_p$ controls the theory, with extensive examples from Klein four and elementary abelian $p$-groups. The work connects dualities with inner products, studies left/right duals, and sets the stage for dualities in finite rings and more general module alphabets, offering concrete tools for coding-theoretic applications and future algebraic extensions.
Abstract
The choice of an isomorphism, a duality, between a finite abelian group $A$ and its character group allows one to define dual codes of additive codes over $A$. Properties of dualities and dual codes are studied, continuing work of Delsarte from 1973 and more recent work of Dougherty and his collaborators.