On Galois LCD codes and LCPs of codes over mixed alphabets
Leijo Jose, Anuradha Sharma
TL;DR
This work develops a comprehensive framework for Galois LCD codes and linear complementary pairs over mixed alphabets in the R-checkR setting. It establishes a precise, practically verifiable criterion that a weakly-free R check R code is monomially equivalent to a Galois LCD code when the residue field is large, and provides explicit enumeration formulas for Euclidean and Hermitian LCD codes, enabling classification in concrete block-lengths. The paper also builds a robust theory of LCPs for R check R codes, derives direct-sum masking schemes with provable security thresholds against fault injection and side-channel attacks, and demonstrates applications to the noiseless two-user adder channel. Collectively, these results advance both the algebraic understanding and the practical resilience of codes over mixed alphabets, with potential impact on secure computation and multiuser communication.
Abstract
Let $\mathtt{R}$ be a finite commutative chain ring with the maximal ideal $γ\mathtt{R}$ of nilpotency index $e\geq 2,$ and let $\check{\mathtt{R}}=\mathtt{R}/γ^{s}\mathtt{R}$ for some positive integer $ s< e.$ In this paper, we study and characterize Galois $\mathtt{R}\check{\mathtt{R}}$-LCD codes of an arbitrary block-length. We show that each weakly-free $\mathtt{R}\check{\mathtt{R}}$-linear code is monomially equivalent to a Galois $\mathtt{R}\check{\mathtt{R}}$-LCD code when $|\mathtt{R}/γ\mathtt{R}|>4,$ while it is monomially equivalent to a Euclidean $\mathtt{R}\check{\mathtt{R}}$-LCD code when $|\mathtt{R}/γ\mathtt{R}|>3.$ We also obtain enumeration formulae for all Euclidean and Hermitian $\mathtt{R}\check{\mathtt{R}}$-LCD codes of an arbitrary block-length. With the help of these enumeration formulae, we classify all Euclidean $\mathbb{Z}_4 \mathbb{Z}_{2}$-LCD codes and $\mathbb{Z}_9 \mathbb{Z}_{3}$-LCD codes of block-lengths $(1,1),$ $(1,2),$ $(2,1),$ $(2,2),$ $(3,1)$ and $(3,2)$ and all Hermitian $\frac{\mathbb{F}_{4}[u]}{\langle u^2\rangle} \;\mathbb{F}_{4}$-LCD codes of block-lengths $(1,1),$ $(1,2),$ $(2,1)$ and $(2,2)$ up to monomial equivalence. Apart from this, we study and characterize LCPs of $\mathtt{R}\check{\mathtt{R}}$-linear codes. We further study a direct sum masking scheme constructed using LCPs of $\mathtt{R}\check{\mathtt{R}}$-linear codes and obtain its security threshold against fault injection and side-channel attacks. We also discuss another application of LCPs of $\mathtt{R}\check{\mathtt{R}}$-linear codes in coding for the noiseless two-user adder channel.