On Galois LCD codes and LCPs of codes over mixed alphabets

Abstract

Let R be a finite commutative chain ring with the maximal ideal $\gamma R$ of nilpotency index $e\ge 2$, and let $\check{R}=R/{\gamma }^{s}R$ for some positive integer $s<e$. In this paper, we study and characterize Galois $R\check{R}$-LCD codes of an arbitrary block-length. We show that each weakly-free $R\check{R}$-linear code is monomially equivalent to a Galois $R\check{R}$-LCD code when $|R/\gamma R|>4$, while it is monomially equivalent to a Euclidean $R\check{R}$-LCD code when $|R/\gamma R|>3$. We also obtain enumeration formulae for all Euclidean and Hermitian $R\check{R}$-LCD codes of an arbitrary block-length. With the help of these enumeration formulae, we classify all Euclidean $Z_4{Z}_2$-LCD codes and $Z_9{Z}_3$-LCD codes of block-lengths $(1,1)$, $(1,2)$, $(2,1)$, $(2,2)$, $(3,1)$ and $(3,2)$ and all Hermitian $\frac{F_4\left[u\right]}{〈u^2〉}{F}_4$-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 $R\check{R}$-linear codes. We further study a direct sum masking scheme constructed using LCPs of $R\check{R}$-linear codes and obtain its security threshold against fault injection and side-channel attacks. We also discuss another application of LCPs of $R\check{R}$-linear codes in coding for the noiseless two-user adder channel.