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How Many Weights Can a Quasi-Cyclic Code Have? | IEEE Journals & Magazine | IEEE Xplore

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, ℓ, k, q), that is defined to be the largest numb...Show More

Abstract:

We investigate the largest number of nonzero weights of quasi-cyclic codes. In particular, we focus on the function ΓQ(n, ℓ, k, q), that is defined to be the largest number of nonzero weights a quasi-cyclic code of index gcd(ℓ, 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.
Published in: IEEE Transactions on Information Theory ( Volume: 66, Issue: 11, November 2020)
Page(s): 6855 - 6862
Date of Publication: 12 June 2020

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