Applications of Finite Frobenius Rings to the Foundations of Algebraic Coding Theory (original) (raw)

The linear programming bound for codes over finite Frobenius rings

Designs, Codes and Cryptography, 2007

In traditional algebraic coding theory the linear-programming bound is one of the most powerful and restrictive bounds for the existence of both linear and non-linear codes. This article develops a linear-programming bound for block codes on finite Frobenius rings. Keywords Codes over rings • Finite Frobenius rings • Homogeneous weights • Linear-programming bound AMS Classifications 94B65 • 94B99 • 16L60 Communicated by S. Gao.

Some remarks on non projective Frobenius algebras and linear codes

Designs, Codes and Cryptography

With a small suitable modification, dropping the projectivity condition, we extend the notion of a Frobenius algebra to grant that a Frobenius algebra over a Frobenius commutative ring is itself a Frobenius ring. The modification introduced here also allows Frobenius finite rings to be precisely those rings which are Frobenius finite algebras over their characteristic subrings. From the perspective of linear codes, our work expands one's options to construct new finite Frobenius rings from old ones. We close with a discussion of generalized versions of the MacWilliams identities that may be obtained in this context.

On the Equivalence of Codes over Finite Rings

Applicable Algebra in Engineering, Communication and Computing, 2004

It is known that if a finite ring R is Frobenius then equivalences of linear codes over R are always monomial transformations. Among other results, in this paper we show that the converse of this result holds for finite local and homogeneous semilocal rings. Namely, it is shown that for every finite ring R which is a direct sum of local and homogeneous semilocal subrings, if every Hamming-weight preserving R-linear transformation of a codeC 1 onto a code C 2 is a monomial transformation then R is a Frobenius ring.

On the equivalence of codes over rings and modules

Finite Fields and Their Applications, 2004

In light of the result by Wood that codes over every finite Frobenius ring satisfy a version of the MacWilliams equivalence theorem, a proof for the converse is considered. A strategy is proposed that would reduce the question to problems dealing only with matrices over finite fields. Using this strategy, it is shown, among other things, that any left MacWilliams basic ring is Frobenius. The results and techniques in the paper also apply to related problems dealing with codes over modules. r 2004 Elsevier Inc. All rights reserved.

On bounds for codes over Frobenius rings under homogeneous weights

Discrete Mathematics, 2004

Homogeneous weight functions were introduced by Heise and Constantinescu (Lineare Codes über Restklassenringen ganzer Zahlen und ihre Automorphismen bezüglich einer verallgemeinerten Hamming-Metrik, Ph.D. Thesis, Technische Universität München, 1995; Problemy Peredachi Informatsii 33(3) (1997) 22-28). They appear as a natural generalization of the Hamming weight on finite fields and the Lee weight on Z 4 and have proven to be important in further papers (J. Combin. Theory 92 (2000) 17-28). This article develops a Plotkin and an Elias bound for (not necessarily linear) block codes on finite Frobenius rings that are equipped with this weight.

Finite Quasi-Frobenius Modules and Linear Codes

Journal of Algebra and Its Applications, 2004

The theory of linear codes over finite fields has been extended by A. Nechaev to codes over quasi-Frobenius modules over commutative rings, and by J. Wood to codes over (not necessarily commutative) finite Frobenius rings. In the present paper we subsume these results by studying linear codes over quasi-Frobenius and Frobenius modules over any finite ring. Using the character module of the ring as alphabet we show that fundamental results like MacWilliams' theorems on weight enumerators and code isometry can be obtained in this general setting. * The author thanks DFG and the University of Düsseldorf for support and hospitality.

Linear Codes over Finite Rings

TEMA - Tendências em Matemática Aplicada e Computacional, 2005

In this paper we present a construction technique of cyclic, BCH, alternat, Goppa and Srivastava codes over a local finite commutative rings with identity.

Do non-free LCD codes over finite commutative Frobenius rings exist?

Designs, Codes and Cryptography, 2020

In this paper, we clarify some aspects on LCD codes in the literature. We first prove that a non-free LCD code does not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD code over finite commutative Frobenius rings. We later show that a free constacyclic code over finite chain ring is LCD if and only if it is reversible, and also provide a necessary and sufficient condition for a constacyclic code to be reversible over finite chain rings. We illustrate the minimum Lee-distance of LCD codes over some finite commutative chain rings and demonstrate the results with examples. We also got some new optimal 4 codes of different lengths which are cyclic LCD codes over 4 .

Pontrjagin Duality and Codes over Finite Commutative Rings

We present linear codes over finite commutative rings which are not necessarily Frobenius. We treat the notion of syndrome decoding by using Pontrjagin duality. We also give a version of Delsarte's theorem over rings relating trace codes and subring subcodes.

Codes over affine algebras with a finite commutative chain coefficient ring

Finite Fields and Their Applications, 2018

We consider codes defined over an affine algebra A = R[X1,. .. , Xr]/ t1(X1),. .. , tr(Xr) , where ti(Xi) is a monic univariate polynomial over a finite commutative chain ring R. Namely, we study the A−submodules of 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.