Weights on finite fields and failures of the MacWilliams identities

TL;DR

This paper explores weights on finite fields where the classical MacWilliams identities fail, showing that identical code weight enumerators do not guarantee identical dual code weight enumerators.

Contribution

It identifies and analyzes classes of weights on finite fields that violate the MacWilliams identities, contrasting with the classical Hamming weight case.

Findings

Existence of weight functions with non-MacWilliams behavior

Examples of codes with identical weights but different dual weights

Insights into the limitations of weight enumerator duality

Abstract

In the 1960s, MacWilliams proved that the Hamming weight enumerator of a linear code over a finite field completely determines, and is determined by, the Hamming weight enumerator of its dual code. In particular, if two linear codes have the same Hamming weight enumerator, then their dual codes have the same Hamming weight enumerator. In contrast, there is a wide class of weights on finite fields whose weight enumerators have the opposite behavior: there exist two linear codes having the same weight enumerator, but their dual codes have different weight enumerators.