On the permutation automorphisms of binary cubic codes

TL;DR

The paper investigates binary cubic codes, defined via a fixed-point-free permutation $\sigma$ of order $3$ in the permutation automorphism group $PAut(C)$, and uses the Huffman decomposition $C=F_{\sigma}(C)\oplus E_{\sigma}(C)$ to analyze their structure. It establishes tight dimension and length constraints: for $n=3m\ge 30$, a cubic code with $|PAut(C)|=3$ must have $k\ge 6$, whereas for $n=3m<30$ and $k\le 4$ the automorphism group cannot be of order $3$. The authors develop a framework for higher-dimensional cubic codes, introducing a block-structure approach to $E_{\sigma}(C)$ and two key hypotheses on the M-block, which either force additional automorphisms or yield explicit order-$3$ examples; several higher-dimensional constructions are provided. Finally, the work discusses implications for the putative extremal self-dual $[72,36,16]$ code, proposing conjectures about the relation between $PAut(C)$ and $\sigma$-weight distinctness, supported by computational evidence from smaller cubic codes.

Abstract

A binary linear code whose permutation automorphism group has a fixed point free permutation of order $3$ is called a binary cubic code. The scope of this paper is to investigate the structural properties of binary cubic codes. Let $C$ be a binary cubic $[n,k]$ code. In this paper, we prove that if $n\geq 30$ and $C$ has permutation automorphism group of order three, then $k\geq 6$. Additionally, we show that if $n < 30$ and $k\leq 4$, then the permutation automorphism group of $C$ has order greater than three. Moreover, along the way, we provide some results on the structure of the higher dimensional cubic codes. In particular, we present some results concerning the structure of the putative extremal self-dual $[72,36,16]$ code under the assumption that it is cubic.

Theorems & Definitions (66)

• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 2.1
• Lemma 2.2
• Example 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof