New binary linear codes from quasi-cyclic codes and an augmentation algorithm

In this paper, one-generator binary quasi-cyclic (QC) codes are explored by statistical tools derived from design of experiments. A connection between a structured cyclic class of statistical designs, k-circulant supersaturated designs and QC codes is given. The mathematical structure of the later codes is explored and a link between complementary dual binary QC codes and E(s2)-optimal k-circulant supersaturated designs is established. Moreover, binary QC codes of rate 1/3, 1/4, 1/5, 1/6 and 1/7 are found by utilizing a genetic algorithm. Our approach is based on a search for good or best codes that attain the current best-known lower bounds on the minimum distance of linear codes, formulated as a combinatorial optimization problem. Surveying previous results, it is shown, that our codes reach the current best-known lower bounds on the minimum distance of linear codes with the same parameters.

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

International Journal of Quantum Information ◽

10.1142/s021974991650012x ◽

2016 ◽

Vol 14 (01) ◽

pp. 1650012 ◽

Cited By ~ 12

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].

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The structure of ℤ2ℤ2s-additive cyclic codes

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830918500489 ◽

2018 ◽

Vol 10 (04) ◽

pp. 1850048 ◽

Cited By ~ 3

Author(s):

Ismail Aydogdu ◽

Taher Abualrub

Keyword(s):

Linear Codes ◽

Cyclic Code ◽

Cyclic Codes ◽

Optimal Parameter ◽

Binary Codes ◽

Binary Linear Codes ◽

Additive Codes

[Formula: see text]-additive codes for any integer [Formula: see text] are considered as codes over mixed alphabets. They are a generalization of binary linear codes and linear codes over [Formula: see text] In this paper, we are interested in studying [Formula: see text]-additive cyclic codes. We will give the generator polynomials of these codes. We will also give the minimal spanning sets for these codes. We will define separable [Formula: see text]-additive codes and provide conditions on the generator polynomials for a [Formula: see text]-additive cyclic code to be separable. Finally, we present some examples of optimal parameter binary codes obtained as images of [Formula: see text]-additive cyclic codes.

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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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Granular Gain of Low-Dimensional Lattices from Binary Linear Codes

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences ◽

10.1587/transfun.e95.a.2168 ◽

2012 ◽

Vol E95.A (12) ◽

pp. 2168-2170

Author(s):

Misako KOTANI ◽

Shingo KAWAMOTO ◽

Motohiko ISAKA

Keyword(s):

Linear Codes ◽

Binary Linear Codes ◽

Low Dimensional

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A class of constacyclic codes over the ring Z4[u,v]/ and their Gray images

Filomat ◽

10.2298/fil1908237i ◽

2019 ◽

Vol 33 (8) ◽

pp. 2237-2248 ◽

Cited By ~ 1

Author(s):

Habibul Islam ◽

Om Prakash

Keyword(s):

Cyclic Codes ◽

Constacyclic Codes ◽

Gray Maps ◽

Permutation Equivalent ◽

Quasi Cyclic Codes

In this paper, we study (1 + 2u + 2v)-constacyclic and skew (1 + 2u + 2v)-constacyclic codes over the ring Z4 + uZ4 + vZ4 + uvZ4 where u2 = v2 = 0,uv = vu. We define some new Gray maps and show that the Gray images of (1 + 2u + 2v)-constacyclic and skew (1 + 2u + 2v)-constacyclic codes are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over Z4. Further, we determine the structure of (1 + 2u + 2v)-constacyclic codes of odd length n.

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