Journal ArticleDOI
On ℤprℤps-additive codes
Ismail Aydogdu,Irfan Siap +1 more
TLDR
The algebraic structure of -additive codes which are -submodules where is prime, and and are positive integers is studied, leading to the identification of the type and the cardinalities of these codes.Abstract:
In this paper, we study the algebraic structure of -additive codes which are -submodules where is prime, and and are positive integers. -additive codes naturally generalize and -additive codes which have been introduced recently. The results obtained in this work generalize a great amount of the studies done on additive codes. Especially, we determine the standard forms of generator and parity-check matrices for this family of codes. This leads to the identification of the type and the cardinalities of these codes. Furthermore, we present some bounds on the minimum distance and examples that attain these bounds. Finally, we present some illustrative examples for some special cases.read more
Citations
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Journal ArticleDOI
On double cyclic codes over Z 4
TL;DR: In this article, the generator polynomials of a double cyclic code of length (r, s ) over R are determined as R x -submodules of R x / ( x r -1 )? R x/ ( x s - 1 ).
Journal ArticleDOI
$${\pmb {{\mathbb {Z}}}}_p{\pmb {{\mathbb {Z}}}}_p[v]$$ Z p Z p [ v ] -additive cyclic codes are asymptotically good
Xiaotong Hou,Jian Gao +1 more
TL;DR: It is proved that, for any positive real number 0 < δ < 1 such that the p -ary entropy at k + l 2 δ is less than 1, the relative minimum distance of the random code is convergent to δ and the rate of therandom code is Convergent to 1k + l, where p, k , l are pairwise coprime positive integers.
Journal ArticleDOI
On the structure of Z2Z2[u3]-linear and cyclic codes
TL;DR: A new family of mixed alphabet codes with algebraic structures of linear and cyclic codes defined over Z 2 Z 2 [ u 3 ] -linear codes is introduced and some examples of optimal binary codes are presented through Gray images of Z 1 Z 1 [ u 2 ] -cyclic codes.
Journal ArticleDOI
$${\mathbb {F}}_qR$$-linear skew constacyclic codes and their application of constructing quantum codes
TL;DR: This paper gives the definition of linear skew constacyclic codes over F, and defines a Gray map from F to R preserving the Hermitian orthogonality, and obtains some good quantum error-correcting codes.
Journal ArticleDOI
Asymptotically good ZprZps-additive cyclic codes
Ting Yao,Shixin Zhu,Xiaoshan Kai +2 more
TL;DR: It is proved that there exist numerous asymptotically good Z p r Z p s -additive cyclic codes.
References
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Journal ArticleDOI
The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes
TL;DR: Certain notorious nonlinear binary codes contain more codewords than any known linear code and can be very simply constructed as binary images under the Gray map of linear codes over Z/sub 4/, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes).
Posted Content
The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes
TL;DR: In this paper, it was shown that all the nonlinear binary codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals can be constructed as binary images under the Gray map of linear codes over Z_4, the integers mod 4.
Journal ArticleDOI
Maximum distance q -nary codes
TL;DR: Examples and construction methods are given to show that these codes exist for a number of values of q, k, and r and with the restriction that the codes be linear.
Journal ArticleDOI
Association schemes and coding theory
P. Delsarte,V.I. Levenshtein +1 more
TL;DR: This paper contains a survey of association scheme theory (with its algebraic and analytical aspects) and of its applications to coding theory (in a wide sense); some "universal bounds" are derived for codes and designs in polynomial type association schemes.