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 $e\geq 2,$ and let $\check{\mathtt{R}}=\mathtt{R}/\gamma^{s}\mathtt{R}$ 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 $|\mathtt{R}/\gamma\mathtt{R}|>4,$ while it is monomially equivalent to a Euclidean $\mathtt{R}\check{\mathtt{R}}$-LCD code when $|\mathtt{R}/\gamma\mathtt{R}|>3.$ 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.