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

Filomat ◽  
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.

scholarly journals Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉

2018 ◽  
Vol 16 (1) ◽  
pp. 490-497
Author(s):  
Xiying Zheng ◽  
Bo Kong
Keyword(s):  
Linear Codes ◽  
Gray Map ◽  
Constacyclic Codes ◽  
Macwilliams Identities

AbstractIn this paper, we study linear codes over ring Rk = 𝔽pm[u1, u2,⋯,uk]/〈$\begin{array}{} u^{2}_{i} \end{array} $ = ui, uiuj = ujui〉 where k ≥ 1 and 1 ≤ i, j ≤ k. We define a Gray map from $\begin{array}{} R_{k}^n\,\,\text{to}\,\,{\mathbb F}_{p^m}^{2^kn} \end{array} $ and give the generator polynomials of constacyclic codes over Rk. We also study the MacWilliams identities of linear codes over Rk.

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.

Gray images of (1 − u)-constacyclic codes over the ring 𝔽pk + u𝔽pk + v𝔽pk + uv𝔽pk

2018 ◽  
Vol 11 (02) ◽  
pp. 1850026
Author(s):  
Pramod Kumar Kewat ◽  
Sarika Kushwaha
Keyword(s):  
Positive Integer ◽  
Linear Code ◽  
Linear Codes ◽  
Gray Map ◽  
Constacyclic Codes ◽  
Arbitrary Length ◽  
Optimal Linear ◽  
Optimal Linear Codes

Let [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] is a prime and [Formula: see text] is a positive integer. We define a gray map from a linear code of length [Formula: see text] over the ring [Formula: see text] to a linear code of length [Formula: see text] over the field [Formula: see text]. In this paper, we characterize the gray images of [Formula: see text]-constacyclic codes of an arbitrary length over the ring [Formula: see text] in terms of quasicyclic codes over [Formula: see text]. We obtain some optimal linear codes over [Formula: see text] as gray images.

On the linear codes over the ring Rp

2016 ◽  
Vol 08 (02) ◽  
pp. 1650036 ◽  
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.

scholarly journals A class of three-weight linear codes and their complete weight enumerators

2016 ◽  
Vol 9 (1) ◽  
pp. 133-149 ◽  
Author(s):  
Shudi Yang ◽  
Zheng-An Yao ◽  
Chang-An Zhao
Keyword(s):  
Linear Codes ◽  
Weight Enumerators

Complete weight enumerators of some linear codes and their applications

2015 ◽  
Vol 81 (1) ◽  
pp. 153-168 ◽  
Author(s):  
Chengju Li ◽  
Sunghan Bae ◽  
Jaehyun Ahn ◽  
Shudi Yang ◽  
Zheng-An Yao
Keyword(s):  
Linear Codes ◽  
Weight Enumerators

Improved binary codes and sequence families from Z/sub 4/-linear codes

1996 ◽  
Vol 42 (5) ◽  
pp. 1582-1587 ◽  
Author(s):  
A.G. Shanbhag ◽  
P. Vijay Kumar ◽  
T. Hellesath
Keyword(s):  
Linear Codes ◽  
Binary Codes

scholarly journals A class of linear codes and their complete weight enumerators

2021 ◽  
Vol 15 (1) ◽  
pp. 73-97
Author(s):  
Dandan Wang ◽  
◽  
Xiwang Cao ◽  
Gaojun Luo ◽  
Keyword(s):  
Linear Codes ◽  
Weight Enumerators
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