A Note on the Shift Bound for Cyclic Codes by the DFT

Junru ZHENG
Takayasu KAIDA

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E93-A    No.11    pp.1918-1922
Publication Date: 2010/11/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E93.A.1918
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
Category: Coding Theory
Keyword: 
Blahut theorem,  lower bound,  discrete Fourier transform,  cyclic code,  minimum distance,  

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Summary: 
For cyclic codes some well-known lower bounds and some decoding methods up to the half of the bounds are suggested. Particularly, the shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. In this paper we consider cyclic codes defined by their defining set, and new simple derivation of the shift bound using the discrete Fourier transform with unknown elements and the Blahut theorem is shown. Moreover two examples of binary cyclic codes are given.