2025/10/07 by Monika Yadav, Yadav, Monika, Anuradha Sharma +1 · 1 citation
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #graph theory and CDMA systems #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2510.06069
Let Re,m be a finite commutative chain ring of even characteristic with maximal ideal ⟨ u ⟩ of nilpotency index e ≥ 2, Teichmuller set Tm, and residue field Re,m/⟨ u ⟩ of order 2m. Suppose that 2 ∈ ⟨ uκ⟩ ∖ ⟨ uκ+1⟩ for some even positive integer κ≤ e. In this paper, we provide a recursive method to construct a self-orthogonal code Ce of type \λ1, λ2, …, λe\ and length n over Re,m from a chain D(1)⊆ D(2) ⊆ ⋯ ⊆ D(\lceil (e)/(2) \rceil) of self-orthogonal codes of length n over Tm, and vice versa, where dim D(i)=λ1+λ2+⋯+λi for 1 ≤ i ≤ \lceil (e)/(2) \rceil, the codes D(\lfloor (e+1)/(2) \rfloor-κ),D(\lfloor (e+1)/(2) \rfloor -κ+1),…,D^(\lfloor (e)/(2)\rfloor-\lfloor \fracκ2 \rfloor) satisfy certain additional conditions, and λ1,λ2,…,λe are non-negative integers satisfying 2λ1+2λ2+⋯+2λe-i+1+λe-i+2+λe-i+3+⋯+λi ≤ n for \lceil (e+1)/(2) \rceil ≤ i≤ e. This construction guarantees that Tori(Ce)=D(i) for 1 ≤ i ≤ \lceil (e)/(2) \rceil. By employing this recursive construction method, together with the results from group theory and finite geometry, we derive explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over Re,m. We also demonstrate these results through examples.