Type: Article

More on Codes Over Finite Quotients of Polynomial Rings

Journal: IEEE Access (21693536)Year: 2025Volume: 13Issue: Pages: 15339 - 15345

Kadhim Al-Lami E.Sobhani R.^aAbdollahi A.^aBagherian J.^a

a :University of Isfahan - IRAN(IR) - Isfahan

GoldDOI:10.1109/ACCESS.2025.3531644Language: English

Abstract

Let q=pr be a prime power, Fq be the finite field of order q and f(x) be a monic polynomial in Fq [x]. Set A:=Fq [x]/ < f(x) >. In this paper we continue the study (started by T. P. Berger and N. El Amrani) of A -codes of length l over A , i.e. A -submodules of Al. We introduce two types of unique generating sets, called type I and type II basis of divisors, for an A -code. Using this, we present a building-up construction so that one can obtain all distinct A -codes of length l, with their basis of divisors. We complete the classification for the special case l=2 and enumerate all the A -codes of length 2. As an example, we list all binary index-2 quasi-cyclic codes of lengths 16 and 32, and all ternary index-2 quasi-cyclic codes of lengths 6 and 18, which are best-known codes. © 2025 The Authors.

Author Keywords

basis of divisorsgenerator matrixparity check matrixpolynomial quotient ringPolynomial ring

Other Keywords

Binary codesBuilding codesPolynomial approximationTurbo codesBase of divisorFinite fieldsGenerator matrixParity check matricesPolynomial quotient ringPolynomial quotientsPolynomial ringsPowerQuasicyclic codesSubmodulesMatrix algebra