More on Codes over Finite Quotients of Polynomial Rings

Let q = p r be a prime power, F q be the finite field of order q and f ( x ) be a monic polynomial in F q [ x ]. Set A := F q [ 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 A l . We introduce two ty...

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Bibliographic Details
Published in:IEEE access 2025-01, Vol.13, p.1-1
Main Authors: Al-Lami, E. K., Sobhani, R., Abdollahi, A., Bagherian, J.
Format: Article
Language:English
Subjects:
Basis of divisors
Building codes
Codes
Fields (mathematics)
Generator matrix
Generators
Indexes
Linear codes
Object recognition
Parity check codes
Parity check matrix
Polynomial quotient ring
Polynomial ring
Polynomials
Ring generators
Rings (mathematics)
Vectors
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Summary:Let q = p r be a prime power, F q be the finite field of order q and f ( x ) be a monic polynomial in F q [ x ]. Set A := F q [ 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 A l . 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.
ISSN:2169-3536
2169-3536
DOI:10.1109/ACCESS.2025.3531644