On equivalence of cyclic and dihedral zero-divisor codes having nilpotents of nilpotency degree two as generators

Abstract

Zero-divisor codes are codes constructed using group rings where their generators are zero-divisors. Generally, zero-divisor codes can be equivalent despite their associated groups are non-isomorphic, leading to the proposed conjecture “Every dihedral zero-divisor code has an equivalent form of cyclic zero-divisor code”. This paper is devoted to study equivalence of zero-divisor codes in F_2G having generators from the 2-nilradical of F_2G, consisting of all nilpotents of nilpotency degree 2 of F_2G. Essentially, algebraic structures of 2-nilradicals are first studied in general for both commutative and non-commutative F_2G before specialized into the case when G is cyclic and dihedral. Then, results are used to study the conjecture above in the cases where the codes generators are from their respective 2-nilradicals.