Linear Complementary Equi-Dual Codes

TL;DR

The paper investigates linear complementary equi-dual (LCED) codes, asking when a code $C$ can be paired with a code $D$ permutation equivalent to $C^{\perp}$ so that $F^n=C\oplus D$. It provides a practical LCED criterion based on a generator matrix $G$ and a permutation $P$ with $GPG^{t}$ invertible, and analyzes the standard form $G=(I_k|A)$ to reduce LCED testing to conditions on $A$. A central conjecture proposes that if a standard-form generator is not LCED, then row sums vanish and column sums are unity; the authors supply substantial evidence and proofs for several special cases (e.g., $n=k+1$, $k=2$, and $n=k+2$ under Pi$_k$-type assumptions) and derive wide-ranging implications about when all codes are LCED over certain fields or lengths. These results, together with the Pi$_k$ framework and computational checks, advance understanding of LCED codes and suggest algorithms for constructing dual-equivalent partners, with potential cryptographic applications in resisting side-channel and fault attacks.

Abstract

We call a linear code $C$ with length $n$ over a field $F$, a linear complementary equi-dual code, when there exists a linear code $D$ over $F$ such that $D$ is permutation equivalent to $C^\perp$ and $(C,D)$ is a linear complementary pair of codes, that is, $C+ D=F^n$ and $C\cap D=0$. We first state a necessary condition on a code $C$ to be linear complementary equi-dual. Then, we conjecture that this necessary condition is also sufficient and present several statements which support this conjecture.

Theorems & Definitions (73)

• Definition 2.1
• Example 2.2
• Theorem 2.3: LiuL
• Proposition 2.4
• proof
• Definition 2.5
• Example 2.6
• Example 2.7
• Proposition 2.8
• Lemma 2.9