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Optimal three-weight cyclic codes whose duals are also optimal [article]

Gerardo Vega, Félix Hernández
2021 arXiv   pre-print
We then study the dual codes in this new class, and show that they are also optimal cyclic codes of length q+1, dimension q-2, and minimum Hamming distance 4.  ...  Here we present a new class of optimal three-weight cyclic codes of length q+1 and dimension 3 over any finite field F_q, and show that the nonzero weights are q-1, q, and q+1.  ...  3), we present a new class of optimal three-weight cyclic codes of dimension 3 (Theorem 3).  ... 
arXiv:2107.04579v1 fatcat:nvho4k6e2jgjnkngx7mxfe7wam

Page 796 of Mathematical Reviews Vol. , Issue 81B [page]

1981 Mathematical Reviews  
Some new constant weight codes. IEEE Trans. Inform.  ...  The results of MacWilliams are used in deriving the formulae. Bal Kishan Dass (Delhi) Tairyan, V. I. [Tairjan, V. 1. Weight spectra of some classes of binary cyclic codes.  ... 

Optimal Sets of Frequency Hopping Sequences From Linear Cyclic Codes

Cunsheng Ding, Yang Yang, Xiaohu Tang
2010 IEEE Transactions on Information Theory  
Optimal sets of frequency hopping sequences are constructed with a class of irreducible cyclic codes. As a byproduct, the weight distribution of a subclass of irreducible cyclic codes is determined.  ...  New optimal parameters of sets of frequency hopping sequences are obtained with subcodes of the Reed-Solomon codes.  ...  Parker, for their comments that improved the presentation of this paper.  ... 
doi:10.1109/tit.2010.2048504 fatcat:g25yx4r3trhvxktvqspwansvxi

A new two-weight code and strongly regular graph

T.A. Gulliver
1996 Applied Mathematics Letters  
It is well known that two-weight codes result in strongly regular graphs if the code is projective. In this paper a (154,6,99) quasi-cyclic two-weight code over OF (3) is presented.  ...  Defining polynomials which belong to the same cyclic class are said to be equivalent. For the codes considered in this paper, the number of classes is 33.  ...  A code is called quasi-cyclic (QC) if there is some integer p, such that every cyclic shift of a codeword by p places is again a codeword [1] . If p = 1, the codes are cyclic.  ... 
doi:10.1016/0893-9659(96)00004-3 fatcat:ufmzjsdsdzhazcpcffpaan5esq

Cyclotomic Constructions of Cyclic Codes With Length Being the Product of Two Primes

Cunsheng Ding
2012 IEEE Transactions on Information Theory  
Bounds on their minimum odd-like weight are also proved. Some of the codes presented in this paper are among the best cyclic codes.  ...  In this paper, three types of generalized cyclotomy of order two are described and three classes of cyclic codes of length and dimension are presented and analyzed, where and are two distinct primes.  ...  Mario Blaum for their comments and suggestions that improved the quality of this paper. His research fields are cryptography and coding theory.  ... 
doi:10.1109/tit.2011.2176915 fatcat:vkbyloibvbg2zcudiwk7bmjqp4

Some results on the weight structure of cyclic codes of composite length

C. Hartmann, Tai-Yang Hwang
1976 IEEE Transactions on Information Theory  
The actual minimum distances of some classes of binary cyclic codes of composite length are derived. For other classes new lower bounds on the minimum distance are obtained.  ...  In this work we investigate the weight structure of cyclic codes of composite length n = n l n 2 , where n 1 and n 2 are relatively prime.  ...  Acknowledgment The assistance of Mrs. Ruth Turnpaugh in the typing of the manuscript is gratefully acknowledged.  ... 
doi:10.1109/tit.1976.1055555 fatcat:a5vdelkyrvf37kc4gv7huhsltm

Two new optimal ternary two-weight codes and strongly regular graphs

T.Aaron Gulliver
1996 Discrete Mathematics  
In this paper optimal (84,6,54) and (98,6,63) quasi-cyclic two-weight codes over GF (3) are presented.  ...  Based upon the code parameters, the existence of two strongly regular graphs is established.  ...  Andries Brouwer for bringing this connection between codes and graphs to his attention.  ... 
doi:10.1016/0012-365x(94)00264-j fatcat:zjmgx6i54vee3alfscsto7kcza

Weight distributions of a class of cyclic codes from $\Bbb F_l$-conjugates

Chengju Li, Qin Yue, Ziling Heng
2015 Advances in Mathematics of Communications  
Moreover, we will obtain a class of two-weight cyclic codes and a class of new three-weight cyclic codes. 2010 Mathematics Subject Classification: Primary: 11T71, 11T24; Secondary: 94B15.  ...  In this paper, we shall present a method to determine the weight distribution of the cyclic code C in two cases: (1) gcd( q k −1 l−1 , N ) = 1; (2) l = 2 and f = 2.  ...  Acknowledgements The authors are very grateful to the editor and the anonymous reviewers for their valuable comments and suggestions that improved the quality of this paper.  ... 
doi:10.3934/amc.2015.9.341 fatcat:26xmlzs4yjcynncoftygwefmq4

Page 10582 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
Summary: “We show that the dual code of the binary cyclic code of length 2” —1 with two zeros a,a“ cannot have three weights in the case that m is even and d =0 (mod 3).  ...  The class of quasi-cyclic codes has proved to be a good source for such codes. In this paper, we use the algebraic structure of quasi- cyclic codes and the BCH type bound introduced in [K.  ... 

Page 2255 of Mathematical Reviews Vol. , Issue 83e [page]

1983 Mathematical Reviews  
The number of vectors of weight 2 in the dual code of an irreducible cyclic codes is computed, and it is shown that projected irreducible cyclic codes exist.  ...  Author’s summary: “In this paper a new class of block concatenated matrix codes is investigated.  ... 

The Weight Enumerator of a Class of Cyclic Codes

Changli Ma, Liwei Zeng, Yang Liu, Dengguo Feng, Cunsheng Ding
2011 IEEE Transactions on Information Theory  
In this paper, the weight distributions of the duals of the cyclic codes with two zeros are settled for a few cases.  ...  Cyclic codes with two zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a few cases.  ...  ACKNOWLEDGMENT The authors are very grateful to the reviewers for their detailed comments and suggestions that much improved the quality of this paper.  ... 
doi:10.1109/tit.2010.2090272 fatcat:jjcky3d6vzavlbsrtrpftn2jum

Page 3063 of Mathematical Reviews Vol. , Issue 83h [page]

1983 Mathematical Reviews  
We present a new class of linear codes, which simplifies the decoding process by adopting a concept of product code and improves the efficiency by mod-2 superimposing a new code on the basic codes.  ...  Authors’ summary: “A table of weight distributions of 198 minimal cyclic codes is presented.”  ... 

New Combinatorial Construction Techniques for Low-Density Parity-Check Codes and Systematic Repeat-Accumulate Codes

Alexander Gruner, Michael Huber
2012 IEEE Transactions on Communications  
Based on specific classes of combinatorial designs, the improved code design focuses on high-rate structured codes with constant column weights 3 and higher.  ...  This paper presents several new construction techniques for low-density parity-check (LDPC) and systematic repeat-accumulate (RA) codes.  ...  ACKNOWLEDGMENT The authors thank the anonymous referees for their careful reading and valuable insights that helped improving the presentation of the paper.  ... 
doi:10.1109/tcomm.2012.070912.110164 fatcat:24apwtjezzegvhgnzveu6mjj7i

Optimal Constant Composition Codes From Zero-Difference Balanced Functions

Cunsheng Ding
2008 IEEE Transactions on Information Theory  
In this section, we introduce a construction of (n; M; d; [w0; . . . ; wq01])q CCCs that meet the Luo-Fu-Vinck-Chen 0018-9448/$25.00  ...  Another Class of Optimal Constant Composition Codes With New Parameters Before describing the class of optimal CCCs with new parameters, we need to introduce cyclotomy.  ...  In addition, two classes of optimal constant composition codes with new parameters are reported.  ... 
doi:10.1109/tit.2008.2006420 fatcat:om7nxbjhsvcevhrmgc7frwceym

A Construction of Binary Punctured Linear Codes and A Supporting Method for Best Code Search

Takuya OHARA, Makoto TAKITA, Masakatu MORII
2021 IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences  
We also propose a method to rapidly derive the minimum weight of the modified cyclic codes.  ...  A cyclic code loses its cyclic structure when it is modified, so we extend the k-sparse algorithm to use it for modified cyclic codes as well.  ...  In cyclic codes, a codeword of weight w always generates a w-class, so if we can generate at most one representative codeword from all the w-class, we can derive A w without verifying all the codewords  ... 
doi:10.1587/transfun.2021tap0007 fatcat:nz4sw7irljasziqukps4a6d62y
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