The weight distributions of cyclic codes with two zeros and zeta functions

We consider the weight distribution of the binary cyclic code of length 2 n − 1 with two zeros α a , α b . Our proof gives information in terms of the zeta function of an associated variety. We carry out an explicit determination of the weight distribution in two cases, for the cyclic codes with zer...

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Bibliographic Details
Published in:Journal of symbolic computation 2010-07, Vol.45 (7), p.723-733
Main Authors: Boston, Nigel, McGuire, Gary
Format: Article
Language:English
Subjects:
Cyclic code
Weight distribution
Zeta function
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Summary:We consider the weight distribution of the binary cyclic code of length 2 n − 1 with two zeros α a , α b . Our proof gives information in terms of the zeta function of an associated variety. We carry out an explicit determination of the weight distribution in two cases, for the cyclic codes with zeros α 3 , α 5 and α , α 11 . These are the smallest cases of two infinite families where finding the weight distribution is an open problem. Finally, an interesting application of our methods is that we can prove that these two codes have the same weight distribution for all odd n .
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2010.03.007