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Weights on finite fields and failures of the MacWilliams identities

Jay A. Wood

TL;DR

This work analyzes generalized weight enumerators on finite fields and shows that duality, as captured by MacWilliams identities, is rare for non-Hamming weights. The authors develop an orbit-based, multiplicity-function framework that reduces the problem to solving linear systems $Woldsymbol heta=oldsymbol extomega$ for two-weight target distributions, then scale to construct two codes with identical $w$-weight enumerators yet different dual weight profiles. A key contribution is a rigorous set of conditions under which one can guarantee non-equality of dual distributions, culminating in a formal theorem (MainV2). The results clarify the limits of duality-respecting weights and provide explicit constructions and examples illustrating when duality fails or holds, with implications for symmetrized enumerators and generalized coding paradigms.

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.

Weights on finite fields and failures of the MacWilliams identities

TL;DR

This work analyzes generalized weight enumerators on finite fields and shows that duality, as captured by MacWilliams identities, is rare for non-Hamming weights. The authors develop an orbit-based, multiplicity-function framework that reduces the problem to solving linear systems for two-weight target distributions, then scale to construct two codes with identical -weight enumerators yet different dual weight profiles. A key contribution is a rigorous set of conditions under which one can guarantee non-equality of dual distributions, culminating in a formal theorem (MainV2). The results clarify the limits of duality-respecting weights and provide explicit constructions and examples illustrating when duality fails or holds, with implications for symmetrized enumerators and generalized coding paradigms.

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.
Paper Structure (9 sections, 20 theorems, 79 equations, 1 figure)

This paper contains 9 sections, 20 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Presenting linear codes via multiplicity functions
  4. Dual codewords of small weight
  5. Some structured matrices
  6. Constructing two-weight codes
  7. Final arguments
  8. Symmetrized enumerators
  9. Examples

Key Result

Theorem 2.8

Let $w$ be an integer-valued weight on $\mathbb{F}_q$, with $w(0)=0$ and $w(r)>0$ for $r \neq 0$. Assume $w$ satisfies: Then, for some $n$, there exist two linear codes $C,D \subseteq \mathbb{F}_q^n$, with $\mathop{\mathrm{wwe}}\nolimits_C = \mathop{\mathrm{wwe}}\nolimits_D$, but $\mathop{\mathrm{wwe}}\nolimits_{C^\perp} \neq \mathop{\mathrm{wwe}}\nolimits_{D^\perp}$. In particular, $w$ does not

Figures (1)

  • Figure 3.1: Representatives of $\mathop{\mathrm{Sym}}\nolimits(w)$-orbits in $\mathbb{F}_q^2$

Theorems & Definitions (54)

  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Remark 2.7
  • Theorem 2.8
  • Example 4.1
  • Proposition 4.2
  • proof
  • Remark 4.3
  • ...and 44 more