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A relation between quasi-cyclic codes and 2-d cyclic codes

FINITE FIELdS ANd THEIR APPLICATIONS 2012年第1期18卷 123-132页

作者： Guneri, Cem Ozbudak, Ferruh Sabanci Univ Fac Engn & Nat Sci TR-34956 Istanbul Turkey Middle E Tech Univ Dept Math TR-06531 Ankara Turkey Middle E Tech Univ Inst Appl Math TR-06531 Ankara Turkey

We consider a q-ary quasi-cyclic code C of length ml and index l, where both in and l are relatively prime to q. If the constituents of C are cyclic codes, we show that C can also be viewed as a 2-d cyclic code of size m x l over F(q). In case in and l are also coprime to each other, we easily observe that the code C must be equivalent to a cyclic code, which was proved earlier by Lim. (C) 2011 Elsevier Inc. All rights reserved.

关键词： Quasi-cyclic code 2-d cyclic code Trace representation

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Artin-Schreier curves and weights of two-dimensional cyclic codes

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FINITE FIELdS ANd THEIR APPLICATIONS 2004年第4期10卷 481-505页

作者： Güneri, C Sabanci Univ Fac Engn & Nat Sci TR-34956 Istanbul Turkey

Let F-q be the finite field with q elements of characteristic p, F-qm be the extension of degree m > 1 and f (x) be a polynomial over F-qm. The maximum number of affine F-qm-rational points that a curve of the form y(q) - y = f(x) can have is q(m+1). We determine a necessary and sufficient condition for such a curve to achieve this maximum number. Then we study the weights of two-dimensional (2-d) cyclic codes. For this, we give a trace representation of the codes starting with the zeros of the dual 2-d cyclic code. This leads to a relation between the weights of codewords and a family of Artin-Schreier curves. We give a lower bound on the minimum distance for a large class of 2-d cyclic codes. Then we look at some special classes that are not covered by our main result and obtain similar minimum distance bounds. (C) 2003 Elsevier Inc. All rights reserved.

关键词： Artin-Schreier curve 2-d cyclic code trace code

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