MacWilliams Type Identity for M-Spotty Rosenbloom-Tsfasman Weight Enumerator of Linear Codes over Finite Ring

2013 ◽  
Vol E96.A (6) ◽  
pp. 1496-1500
Author(s):  
Jianzhang CHEN ◽  
Wenguang LONG ◽  
Bo FU
Keyword(s):  
Linear Codes ◽  
Finite Ring ◽  
Weight Enumerator ◽  
Type Identity

DNA computing over the ring ℤ4[v]/〈v2 − v〉

2018 ◽  
Vol 11 (07) ◽  
pp. 1850090
Author(s):  
Narendra Kumar ◽  
Abhay Kumar Singh
Keyword(s):  
Significant Role ◽  
Dna Computing ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Finite Ring ◽  
Dna Codes ◽  
Reversible Cyclic Codes ◽  
Text Images

In this paper, we discuss the DNA construction of general length over the finite ring [Formula: see text], with [Formula: see text], which plays a very significant role in DNA computing. We discuss the GC weight of DNA codes over [Formula: see text]. Several examples of reversible cyclic codes over [Formula: see text] are provided, whose [Formula: see text]-images are [Formula: see text]-linear codes with good parameters.

Linear codes from vectorial Boolean functions in the context of algebraic attacks

2020 ◽  
pp. 2150032
Author(s):  
M. Boumezbeur ◽  
S. Mesnager ◽  
K. Guenda
Keyword(s):  
Boolean Functions ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Weight Enumerator ◽  
Direct Link ◽  
Algebraic Attacks ◽  
Vectorial Boolean Functions ◽  
Lcd Codes ◽  
Vectorial Function ◽  
The Relationship

In this paper, we study the relationship between vectorial (Boolean) functions and cyclic codes in the context of algebraic attacks. We first derive a direct link between the annihilators of a vectorial function (in univariate form) and certain [Formula: see text]-ary cyclic codes (which we show that they are LCD codes). We also present some properties of those cyclic codes as well as their weight enumerator. In addition, we generalize the so-called algebraic complement and study its properties.

scholarly journals FINITE QUASI-FROBENIUS MODULES AND LINEAR CODES

2004 ◽  
Vol 03 (03) ◽  
pp. 247-272 ◽  
Author(s):  
MARCUS GREFERATH ◽  
ALEXANDR NECHAEV ◽  
ROBERT WISBAUER
Keyword(s):  
Finite Fields ◽  
Linear Codes ◽  
General Setting ◽  
Commutative Rings ◽  
Finite Ring ◽  
Weight Enumerators ◽  
Frobenius Rings ◽  
Finite Frobenius Rings

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.

MACWILLIAMS IDENTITIES FOR WEIGHT ENUMERATORS WITH RESPECT TO THE RT METRIC

2014 ◽  
Vol 06 (02) ◽  
pp. 1450030 ◽  
Author(s):  
AMIT K. SHARMA ◽  
ANURADHA SHARMA
Keyword(s):  
Algebraic Structure ◽  
Linear Code ◽  
Linear Codes ◽  
Error Correcting Codes ◽  
Weight Enumerator ◽  
Weight Enumerators ◽  
Ring Of Integers ◽  
The Past ◽  
Macwilliams Identities ◽  
Hamming 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.

Some results on the linear codes over the finite ring F2+v1F2+⋯+vrF2

2016 ◽  
Vol 14 (01) ◽  
pp. 1650012 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Senol Eren
Keyword(s):  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Finite Ring ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Quantum Error ◽  
Quasi Cyclic Codes

In this paper, we study the structure of cyclic, quasi-cyclic codes and their skew codes over the finite ring [Formula: see text], [Formula: see text] for [Formula: see text]. The Gray images of cyclic, quasi-cyclic, skew cyclic, skew quasi-cyclic codes over [Formula: see text] are obtained. A necessary and sufficient condition for cyclic code over [Formula: see text] that contains its dual has been given. The parameters of quantum error correcting codes are obtained from cyclic codes over [Formula: see text].

scholarly journals A MacWilliams-type identity for linear codes on weak order

2003 ◽  
Vol 262 (1-3) ◽  
pp. 181-194 ◽  
Author(s):  
Dae San Kim ◽  
Jeh Gwon Lee
Keyword(s):  
Linear Codes ◽  
Weak Order ◽  
Type Identity

scholarly journals A new MacWillams type identity for linear codes

1996 ◽  
Vol 25 (3) ◽  
pp. 651-656 ◽  
Author(s):  
Keisuke SHIROMOTO
Keyword(s):  
Linear Codes ◽  
Type Identity

scholarly journals A Characterization of Graphs by Codes from their Incidence Matrices

10.37236/2770 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Peter Dankelmann ◽  
Jennifer D. Key ◽  
Bernardo G. Rodrigues
Keyword(s):  
Incidence Matrix ◽  
Linear Codes ◽  
Minimum Weight ◽  
Weight Enumerator ◽  
Edge Connectivity ◽  
Connected Graphs ◽  
Incidence Matrices ◽  
Wide Range ◽  
The Matrix

We continue our earlier investigation of properties of linear codes generated by the rows of incidence matrices of $k$-regular connected graphs on $n$ vertices. The notion of edge connectivity is used to show that, for a wide range of such graphs, the $p$-ary code, for all primes $p$, from an $n \times \frac{1}{2}nk$ incidence matrix has dimension $n$ or $n-1$, minimum weight $k$, the minimum words are the scalar multiples of the rows, there is a gap in the weight enumerator between $k$ and $2k-2$, and the words of weight $2k-2$ are the scalar multiples of the differences of intersecting rows of the matrix. For such graphs, the graph can thus be retrieved from the code.

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