On bounds for codes over Frobenius rings under homogeneous weights (original) (raw)
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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.
International Journal of Algebra and Statistics, 2019
In this paper, some lower and upper bounds on the covering radius of codes over the nite non chain ring A = F4 + vF4; v2 = v with respect to Bachoc weight is given. Also, the covering radius of various Block Repetition Codes of same and different length over the nite non chain ring A = F4 + vF4; v2 = v is obtained.In this paper, some lower and upper bounds on the covering radius of codes over the nite non chain ring A = F4 + vF4; v2 = v with respect to Bachoc weight is given. Also, the covering radius of various Block Repetition Codes of same and different length over the nite non chain ring A = F4 + vF4; v2 = v is obtained.
Weight enumerators and gray maps of linear codes over rings
2006
The main focus in this thesis is linear codes over rings. In the first part, we look at linear codes over Galois rings, and using the homogeneous weight, we improve upon Wilson's results about the prime power that divides the coefficients of the homogeneous weight enumerators of these codes. We also prove that our results are best possible. Our results about homogeneous weight enumerators of linear codes over Galois rings generalize the results that we have for the Lee weight enumerators of linear codes over the ring of integers modulo 4. We also consider other weight enumerators, in particular the complete weight enumerators of linear codes and we obtain MacWilliams-like identities for these weight enumerators considering different rings. These MacWilliams-like identities lead to MacWilliams identities for the Hamming weight enumerators of linear codes over rings. We also give a counter-example to show that we cannot have MacWilliams-like identities for the Euclidean weight enu...
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.
One-Homogeneous Weight Codes Over Finite Chain Rings
Bulletin of the Korean Mathematical Society, 2015
This paper determines the structures of one-homogeneous weight codes over finite chain rings and studies the algebraic properties of these codes. We present explicit constructions of one-homogeneous weight codes over finite chain rings. By taking advantage of the distancepreserving Gray map defined in [7] from the finite chain ring to its residue field, we obtain a family of optimal one-Hamming weight codes over the residue field. Further, we propose a generalized method that also includes the examples of optimal codes obtained by Shi et al. in [17].
Information and Control, 1972
Given an integer m which is a product of distinct primes Pi, a method is given for constructing codes over the ring of integers modulo m from cyclic codes over GF(pi). Specifically, if we are given a cyclic (n, ki) code over GF(pt) with minimum Hamming distance di, for each i, then we construct a code of block length n over the integers modulo m with 1-[~ p~i codewords, which is both linear and cyclic and has minimum Hamming distance mini di. i j k
The homogeneous weight for Rk, related gray map and new binary quasi-cyclic codes
Filomat, 2017
Using theoretical results about the homogeneous weights for Frobenius rings, we describe the homogeneous weight for the ring family Rk, a recently introduced family of Frobenius rings which have been used extensively in coding theory. We find an associated Gray map for the homogeneous weight using first order Reed-Muller codes and we describe some of the general properties of the images of codes over Rk under this Gray map. We then discuss quasi-twisted codes over Rk and their binary images under the homogeneous Gray map. In this way, we find many optimal binary codes which are self-orthogonal and quasi-cyclic. In particular, we find a substantial number of optimal binary codes that are quasi-cyclic of index 8, 16 and 24, nearly all of which are new additions to the database of quasi-cyclic codes kept by Chen.
On the Classification of Codes over Non-Unital Ring of Order 4
Cornell University - arXiv, 2022
In the last 60 years coding theory has been studied a lot over finite fields Fq or commutative rings R with unity. Although in 1993, a study on the classification of the rings (not necessarily commutative or ring with unity) of order p 2 had been presented, the construction of codes over non-commutative rings or non-commutative non-unital rings surfaced merely two years ago. In this letter, we extend the diverse research on exploring the codes over the non-commutative and non-unital ring E = 2a = 2b = 0, a 2 = a, b 2 = b, ab = a, ba = b by presenting the classification of optimal and nice codes of length n ≤ 7 over E, along-with respective weight enumerators and complete weight enumerators.
Computation of Minimum Hamming Weight for Linear Codes
Iranian Journal of Mathematical Sciences and Informatics, 2019
In this paper, we consider the minimum Hamming weight for linear codes over special finite quasi-Frobenius rings. Furthermore, we obtain minimal free R-submodules of a finite quasi-Frobenius ring R which contain a linear code and derive the relation between their minimum Hamming weights. Finally, we suggest an algorithm that computes this weight using the Gröbner basis and we show that under certain conditions a linear code takes the maximum of minimum Hamming weight.