Codes over Affine Algebras with a Finite Commutative Chain coefficient Ring

2017/09/16 by Edgar Martı́nez-Moro, Martínez-Moro, E., Alejandro P. Nicolás +3
Computer Science · Engineering · #Coding theory and cryptography #graph theory and CDMA systems #Cooperative Communication and Network Coding

paper · pdf · doi:10.48550/arxiv.1709.05464

Abstract

We consider codes defined over an affine algebra \mathcal A=R[X₁,…,X_r]/⟨ t₁(X₁),…,t_r(X_r)⟩, where t_i(X_i) is a monic univariate polynomial over a finite commutative chain ring R. Namely, we study the \mathcal A-submodules of \mathcal A^l (l∈ ℕ). These codes generalize both the codes over finite quotients of polynomial rings and the multivariable codes over finite chain rings. Some codes over Frobenius local rings that are not chain rings are also of this type. A canonical generator matrix for these codes is introduced with the help of the Canonical Generating System. Duality of the codes is also considered.

Related

• Abelian and consta-Abelian polyadic codes over affine algebras with a finite commutative chain coefficient ring
• Codes over an algebra over ring
• Multivariable codes over finite chain rings: semisimple codes
• Primitive Idempotents and Constacyclic Codes over Finite Chain Rings
• Bounds and MacWilliams Identities for codes over Artinian rings