MACWILLIAMS IDENTITIES FOR WEIGHT ENUMERATORS WITH RESPECT TO THE RT METRIC

Linear codes constitute an important family of error-correcting codes and have a rich algebraic structure. Initially, these codes were studied with respect to the Hamming metric; while for the past few years, they are also studied with respect to a non-Hamming metric, known as the Rosenbloom–Tsfasman metric (also known as RT metric or ρ metric). In this paper, we introduce and study the split ρ weight enumerator of a linear code in the R-module Mn×s(R) of all n × s matrices over R, where R is a finite Frobenius commutative ring with unity. We also define the Lee complete ρ weight enumerator of a linear code in Mn×s(ℤk), where ℤk is the ring of integers modulo k ≥ 2. We also derive the MacWilliams identities for each of these ρ weight enumerators.

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Blockwise and Low Density Key Error Correcting Codes

International Journal of Mathematical Engineering and Management Sciences ◽

10.33889/ijmems.2020.5.6.092 ◽

2020 ◽

Vol 5 (6) ◽

pp. 1234-1248

Author(s):

Pankaj Kumar Das ◽

Subodh Kumar

Keyword(s):

Linear Code ◽

Communication System ◽

Linear Codes ◽

Upper Bounds ◽

Error Correcting Codes ◽

Low Density ◽

Code Length ◽

Lower And Upper Bounds ◽

Noisy Channels ◽

Know How

To protect the information from disturbances created by noisy channels, redundant symbols (called check symbols) with the information symbols are added. These extra symbols play important role for the efficiency of the communication system. It is always important to know how much these check symbols are required for a code designed for a specific purpose. In this communication, we give lower and upper bounds on check symbols needed to a linear code correcting key errors of length upto p which are confined to a single sub-block. We provide two examples of such linear codes. We, further, obtain those bounds for the case when key error occurs in the whole code length, but the number of disturbing components within key error is upto a certain number. Two examples in this case also are provided.

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Type IV codes over a non-unital ring

Journal of Algebra and Its Applications ◽

10.1142/s0219498822501420 ◽

2021 ◽

pp. 2250142

Author(s):

Adel Alahmadi ◽

Alaa Altassan ◽

Widyan Basaffar ◽

Hatoon Shoaib ◽

Alexis Bonnecaze ◽

...

Keyword(s):

Local Ring ◽

Algebraic Structure ◽

Invariant Theory ◽

Linear Codes ◽

Hamming Weight ◽

Dual Codes ◽

Weight Enumerators ◽

Type Iv ◽

Commutative Local Ring ◽

Torsion Codes

There is a special local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by [Formula: see text] We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over [Formula: see text] and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

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On the linear codes over the ring Rp

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830916500361 ◽

2016 ◽

Vol 08 (02) ◽

pp. 1650036 ◽

Cited By ~ 2

Author(s):

Abdullah Dertli ◽

Yasemin Cengellenmis ◽

Senol Eren

Keyword(s):

Prime Number ◽

Linear Codes ◽

Cyclic Codes ◽

Error Correcting Codes ◽

Constacyclic Codes ◽

Nontrivial Automorphism ◽

Macwilliams Identities ◽

Skew Cyclic Codes ◽

Quantum Error ◽

Quasi Cyclic Codes

Some results are generalized on linear codes over [Formula: see text] in [15] to the ring [Formula: see text], where [Formula: see text] is an odd prime number. The Gray images of cyclic and quasi-cyclic codes over [Formula: see text] are obtained. The parameters of quantum error correcting codes are obtained from negacyclic codes over [Formula: see text]. A nontrivial automorphism [Formula: see text] on the ring [Formula: see text] is determined. By using this, the skew cyclic, skew quasi-cyclic, skew constacyclic codes over [Formula: see text] are introduced. The number of distinct skew cyclic codes over [Formula: see text] is given. The Gray images of skew codes over [Formula: see text] are obtained. The quasi-constacyclic and skew quasi-constacyclic codes over [Formula: see text] are introduced. MacWilliams identities of linear codes over [Formula: see text] are given.

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MacWilliams Identities and Matroid Polynomials

The Electronic Journal of Combinatorics ◽

10.37236/1636 ◽

2002 ◽

Vol 9 (1) ◽

Cited By ~ 6

Author(s):

Thomas Britz

Keyword(s):

Linear Code ◽

Tutte Polynomial ◽

Decomposition Theorem ◽

Weight Enumerators ◽

Support Weight ◽

Macwilliams Identities

We present generalisations of several MacWilliams type identities, including those by Kløve and Shiromoto, and of the theorems of Greene and Barg that describe how the Tutte polynomial of the vector matroid of a linear code determines the $r$th support weight enumerators of the code. One of our main tools is a generalisation of a decomposition theorem due to Brylawski.

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Linear, cyclic and constacyclic codes over S4 = F2 + uF2 + u2F2 + u3F2

Filomat ◽

10.2298/fil1405897o ◽

2014 ◽

Vol 28 (5) ◽

pp. 897-906

Author(s):

Ödemiş Özger ◽

Ümmü Kara ◽

Bahattin Yıldız

Keyword(s):

Linear Codes ◽

Gray Map ◽

Binary Codes ◽

Constacyclic Codes ◽

Weight Enumerators ◽

Lee Weight ◽

Macwilliams Identities

In this work, linear codes over the ring S4 = F2 + uF2 + u2F2 + u3F2 are considered. The Lee weight and gray map for codes over S4 are defined and MacWilliams identities for the complete, the symmetrized and the Lee weight enumerators are obtained. Cyclic and (1 + u2)-constacyclic codes over S4 are studied, as a result of which a substantial number of optimal binary codes of different lengths are obtained as the Gray images of cyclic and constacyclic codes over S4.

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