113 research outputs found
On the lengths of divisible codes
In this article, the effective lengths of all -divisible linear codes
over with a non-negative integer are determined. For that
purpose, the -adic expansion of an integer is introduced. It is
shown that there exists a -divisible -linear code of
effective length if and only if the leading coefficient of the
-adic expansion of is non-negative. Furthermore, the maximum weight
of a -divisible code of effective length is at most ,
where denotes the cross-sum of the -adic expansion of .
This result has applications in Galois geometries. A recent theorem of
N{\u{a}}stase and Sissokho on the maximum size of a partial spread follows as a
corollary. Furthermore, we get an improvement of the Johnson bound for constant
dimension subspace codes.Comment: 17 pages, typos corrected; the paper was originally named "An
improvement of the Johnson bound for subspace codes
Intersection numbers for subspace designs
Intersection numbers for subspace designs are introduced and -analogs of
the Mendelsohn and K\"ohler equations are given. As an application, we are able
to determine the intersection structure of a putative -analog of the Fano
plane for any prime power . It is shown that its existence implies the
existence of a - subspace design. Furthermore, several
simplified or alternative proofs concerning intersection numbers of ordinary
block designs are discussed
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