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Register a new userClassification of $Delta$-divisible linear codes spanned by codewords of weight $Delta$
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We classify all $q$-ary $Delta$-divisible linear codes which are spanned by codewords of weight $Delta$. The basic building blocks are the simplex codes, and for $q=2$ additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codewords of weight $4$ have been classified, which is the case $q=2$ and $Delta=4$ of our classification. As an application, we give an alternative proof of a theorem of Liu on binary $Delta$-divisible codes of length $4Delta$ in the projective case.
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For which positive integers $n,k,r$ does there exist a linear $[n,k]$ code $C$ over $mathbb{F}_q$ with all codeword weights divisible by $q^r$ and such that the columns of a generating matrix of $C$ are projectively distinct? The motivation for studying this problem comes from the theory of partial spreads, or subspace codes with the highest possible minimum distance, since the set of holes of a partial spread of $r$-flats in $operatorname{PG}(v-1,mathbb{F}_q)$ corresponds to a $q^r$-divisible code with $kleq v$. In this paper we provide an introduction to this problem and report on new results for $q=2$.
The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singletons defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations we are able to derive known identities (in an easier way) for interesting cases, such as extremal codes, Hermitian codes, MDS and NMDS codes. Moreover, we are able to present for the first time the weight distribution of AMDS codes. We also discuss the link between our results and the Pless equations.
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