Additive codes attain the Griesmer bound with equality for sufficiently large minimum distance, giving infinite series of optimal codes superior to linear codes.
Projective divisible binary codes
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abstract
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 $k\leq v$. In this paper we provide an introduction to this problem and report on new results for $q=2$.
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Proposes a canonical augmentation algorithm for classifying linear codes and applies it to obtain results over GF(2), GF(3), and GF(4).
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Additive codes attaining the Griesmer bound
Additive codes attain the Griesmer bound with equality for sufficiently large minimum distance, giving infinite series of optimal codes superior to linear codes.
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Classification of linear codes using canonical augmentation
Proposes a canonical augmentation algorithm for classifying linear codes and applies it to obtain results over GF(2), GF(3), and GF(4).