Title data
Kiermaier, Michael ; Kurz, Sascha:
On the lengths of divisible codes.
Bayreuth
,
2019
.  17 S.
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divisible_codes.pdf  Published Version Available under License Creative Commons BY 4.0: Attribution . Download (205kB) 
Project information
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Project financing: 
Deutsche Forschungsgemeinschaft 
Abstract
In this article, the effective lengths of all q^rdivisible linear codes over GF(q) with a nonnegative integer r are determined. For that purpose, the S_q(r)adic expansion of an integer n is introduced. It is shown that there exists a q^rdivisible GF(q)linear code of effective length n if and only if the leading coefficient of the S_q(r)adic expansion of n is nonnegative. Furthermore, the maximum weight of a q^rdivisible code of effective length n is at most the crosssum of the S_q(r)adic expansion of n. This result has applications in Galois geometries. A recent theorem of Nastase and Sissokho on the maximum sizes of partial spreads follows as a corollary. Furthermore, we get an improvement of the Johnson bound for constant dimension subspace codes.