How many weights can a quasi-cyclic code have ? - Archive ouverte HAL
Journal Articles IEEE Transactions on Information Theory Year : 2020

How many weights can a quasi-cyclic code have ?

Abstract

We investigate the largest number of nonzero weights of quasi-cyclic codes. In particular, we focus on the function ΓQ(n, , k, q), that is defined to be the largest number of nonzero weights a quasi-cyclic code of index gcd(, n), length n and dimension k over Fq can have, and connect it to similar functions related to linear and cyclic codes. We provide several upper and lower bounds on this function, using different techniques and studying its asymptotic behavior. Moreover, we determine the smallest index for which a q-ary Reed-Muller code is quasi-cyclic, a result of independent interest.
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Dates and versions

hal-02790988 , version 1 (05-06-2020)

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Cite

Minjia Shi, Alessandro Neri, Patrick Solé. How many weights can a quasi-cyclic code have ?. IEEE Transactions on Information Theory, 2020, ⟨10.1109/TIT.2020.3001591⟩. ⟨hal-02790988⟩
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