2024/12/13 by Leijo Jose, Jose, Leijo, Anuradha Sharma +1 · 2 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · #Coding theory and cryptography #DNA and Biological Computing #Error Correcting Code Techniques #FOS: Computer and information sciences #Information Theory (cs.IT)
paper · pdf · doi:10.48550/arxiv.2412.09937
openalex publication_date 2024/12/13 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Let mathttR be a finite commutative chain ring with the maximal ideal\n\γ mathttR of nilpotency index e\≥ 2, and let\n check mathttR= mathttR/\γs mathttR for some positive integer\n s< e. In this paper, we study and characterize Galois\n mathttR check mathttR-LCD codes of an arbitrary block-length. We show\nthat each weakly-free mathttR check mathttR-linear code is monomially\nequivalent to a Galois mathttR check mathttR-LCD code when\n| mathttR/\γ mathttR|>4, while it is monomially equivalent to a\nEuclidean mathttR check mathttR-LCD code when\n| mathttR/\γ mathttR|>3. We also obtain enumeration formulae for all\nEuclidean and Hermitian mathttR check mathttR-LCD codes of an\narbitrary block-length. With the help of these enumeration formulae, we\nclassify all Euclidean \ℤ4 \ℤ2-LCD codes and\n\ℤ9 \ℤ3-LCD codes of block-lengths (1,1), (1,2),\n(2,1), (2,2), (3,1) and (3,2) and all Hermitian\n frac mathbbF4[u]\⟨ u2\⟩ ; mathbbF4-LCD codes of\nblock-lengths (1,1), (1,2), (2,1) and (2,2) up to monomial equivalence.\nApart from this, we study and characterize LCPs of\n mathttR check mathttR-linear codes. We further study a direct sum\nmasking scheme constructed using LCPs of mathttR check mathttR-linear\ncodes and obtain its security threshold against fault injection and\nside-channel attacks. We also discuss another application of LCPs of\n mathttR check mathttR-linear codes in coding for the noiseless\ntwo-user adder channel.\n