Journal Article10.1007/S10623-017-0334-8
Z2-double cyclic codes.
TL;DR: The polynomial representation of Z2-double cyclic codes and its duals is given, and the relations between the generator polynomials of these codes are studied.
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Abstract: A binary linear code C is a $${\mathbb {Z}}_2$$
-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the $${\mathbb {Z}}_2[x]$$
-module $${\mathbb {Z}}_2[x]/(x^r-1)\times {\mathbb {Z}}_2[x]/(x^s-1).$$
We determine the structure of $${\mathbb {Z}}_2$$
-double cyclic codes giving the generator polynomials of these codes. We give the polynomial representation of $${\mathbb {Z}}_2$$
-double cyclic codes and its duals, and the relations between the generator polynomials of these codes. Finally, we study the relations between $${{\mathbb {Z}}}_2$$
-double cyclic and other families of cyclic codes, and show some examples of distance optimal $${\mathbb {Z}}_2$$
-double cyclic codes.
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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).
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TL;DR: For binary and quaternary linear codes, the fundamental parameters are found and standard forms for generator and parity-check matrices are given.
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TL;DR: It is shown that the duals of Z2Z4-additive cyclic codes are also cyclic and an infinite family of Maximum Distance separable with respect to the singleton bound codes is presented.
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