# $${{\mathbb {Z}}}_2$$ Z 2 -double cyclic codes

$${{\mathbb {Z}}}_2$$ Z 2 -double cyclic codes A binary linear code C is a $${\mathbb {Z}}_2$$ 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 $${\mathbb {Z}}_2[x]$$ Z 2 [ x ] -module $${\mathbb {Z}}_2[x]/(x^r-1)\times {\mathbb {Z}}_2[x]/(x^s-1).$$ Z 2 [ x ] / ( x r - 1 ) × Z 2 [ x ] / ( x s - 1 ) . We determine the structure of $${\mathbb {Z}}_2$$ Z 2 -double cyclic codes giving the generator polynomials of these codes. We give the polynomial representation of $${\mathbb {Z}}_2$$ Z 2 -double cyclic codes and its duals, and the relations between the generator polynomials of these codes. Finally, we study the relations between $${{\mathbb {Z}}}_2$$ Z 2 -double cyclic and other families of cyclic codes, and show some examples of distance optimal $${\mathbb {Z}}_2$$ Z 2 -double cyclic codes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Designs, Codes and Cryptography Springer Journals

# $${{\mathbb {Z}}}_2$$ Z 2 -double cyclic codes

, Volume 86 (3) – Feb 10, 2017
17 pages

/lp/springer_journal/mathbb-z-2-z-2-double-cyclic-codes-PTFzxHtA0J
Publisher
Springer US
Subject
Mathematics; Combinatorics; Coding and Information Theory; Data Structures, Cryptology and Information Theory; Data Encryption; Discrete Mathematics in Computer Science; Information and Communication, Circuits
ISSN
0925-1022
eISSN
1573-7586
D.O.I.
10.1007/s10623-017-0334-8
Publisher site
See Article on Publisher Site

### Abstract

A binary linear code C is a $${\mathbb {Z}}_2$$ 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 $${\mathbb {Z}}_2[x]$$ Z 2 [ x ] -module $${\mathbb {Z}}_2[x]/(x^r-1)\times {\mathbb {Z}}_2[x]/(x^s-1).$$ Z 2 [ x ] / ( x r - 1 ) × Z 2 [ x ] / ( x s - 1 ) . We determine the structure of $${\mathbb {Z}}_2$$ Z 2 -double cyclic codes giving the generator polynomials of these codes. We give the polynomial representation of $${\mathbb {Z}}_2$$ Z 2 -double cyclic codes and its duals, and the relations between the generator polynomials of these codes. Finally, we study the relations between $${{\mathbb {Z}}}_2$$ Z 2 -double cyclic and other families of cyclic codes, and show some examples of distance optimal $${\mathbb {Z}}_2$$ Z 2 -double cyclic codes.

### Journal

Designs, Codes and CryptographySpringer Journals

Published: Feb 10, 2017

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