Work

A Gröbner approach to dual-containing cyclic left module $ (\theta,\delta) $ -codes over finite commutative Frobenius rings

Year: 2025

Type: article

Abstract: For a skew polynomial ring $ R = A[X;\theta, \delta] $ where $ A $ is a commutative Frobenius ring, $ \theta $ an endomorphism of $ A $ and $ \delta $ a $ \theta $-derivation of $ A $, we consider cyclic left module codes $ \mathcal{C} = Rg/Rf\subset R/Rf $ where $ g $ is a left and right divisor of $ f $ in $ R $. In this paper, we derive a parity check matrix when $ A $ is a finite commutative Frobenius ring using only the framework of skew polynomial rings. We consider rings $ A = B[a_1, \ldots, a_s] $ which are free $ B $-modules where the restriction of $ \delta $ and $ \theta $ to $ B $ are polynomial maps. If a Gröbner basis can be computed over $ B $, then we show that all Euclidean and Hermitian dual-containing codes $ \mathcal{C} = Rg/Rf\subset R/Rf $ can be computed using a Gröbner basis. We also give an algorithm to test if the dual code is again a cyclic left module code. We illustrate our approach for rings of order $ 4 $ with non-trivial endomorphism and the Galois ring of characteristic $ 4 $.

Source: Advances in Mathematics of Communications

DOI: 10.3934/amc.2025019

Authors: Hedongliang Liu, Cornelia Ott, Felix Ulmer

Institutions: Technical University of Munich, Deutsches Zentrum für Luft- und Raumfahrt e. V. (DLR), Centre National de la Recherche Scientifique, Institut de recherche mathématique de Rennes, Université de Rennes

Language: English

Topic: Coding theory and cryptography

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: diamond

Funders: Centre Henri Lebesgue, Deutsche Forschungsgemeinschaft, Agence Nationale de la Recherche

Awards: G1769130723, G2609165258, Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation, G5294259702