On Galois LCD codes and LCPs of codes over mixed alphabets

Let $\mathtt{R}$ be a finite commutative chain ring with the maximal ideal $\gamma \mathtt{R}$ of nilpotency index $e\ge 2,$ and let $\check{\mathtt{R}}=\mathtt{R}\mathrm{/}{\gamma }^s\mathtt{R}$ for some positive integer $s<e\mathrm{.}$ 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 $\mathrm{\mid }\mathtt{R}\mathrm{/}\gamma \mathtt{R}\mathrm{\mid }>4,$ while it is monomially equivalent to a Euclidean $\mathtt{R}\check{\mathtt{R}}$-LCD code when $\mathrm{\mid }\mathtt{R}\mathrm{/}\gamma \mathtt{R}\mathrm{\mid }>3.$ 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 $\frac{{\mathbb{F}}_4[u]}{⟨u^2⟩}{\mathbb{F}}_4$-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.