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Two algorithms for the classification of linear codes over finite fields are presented. One of the algorithms is based on canonical augmentation and the other one on lattice point enumeration. New classification results over fields with 2, 3 and 4 elements are obtained. [ABSTRACT FROM AUTHOR]

In random network coding so-called constant dimension codes (CDCs) are used for error correction and detection. Most of the largest known codes contain a lifted maximum rank distance (LMRD) code as a subset. For some special cases, Etzion and Silberstein have demonstrated that one can obtain tighter upper bounds on the maximum possible cardinality of CDCs if we assume that an LMRD code is contained. The range of applicable parameters was partially extended by Heinlein. Here we fully generalize those bounds, which also sheds some light on recent constructions. [ABSTRACT FROM AUTHOR]

In this article, the effective lengths of all $q^{r}$ -divisible linear codes over $\mathbb {F}_{q}$ with a non-negative 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 $q^{r}$ -divisible $\mathbb {F}_{q}$ -linear code of effective length $n$ if and only if the leading coefficient of the $S_{q}(r)$ -adic expansion of $n$ is non-negative. Furthermore, the maximum weight of a $q^{r}$ -divisible code of effective length $n$ is at most $\sigma q^{r}$ , where $\sigma $ denotes the cross-sum of the $S_{q}(r)$ -adic expansion of $n$. This result has applications in Galois geometries. A recent theorem of Năstase 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. [ABSTRACT FROM AUTHOR]

It is shown that there does not exist a projective triply-even binary code of length 59. This settles the last open length for projective triply-even binary codes, which therefore exist precisely for the lengths 15, 16, 30, 31, 32, 45–51, and ≥ 60. [ABSTRACT FROM AUTHOR]

We show that no projective 16-divisible binary linear code of length 131 exists. This implies several improved upper bounds for constant-dimension codes, used in random linear network coding, and partial spreads. [ABSTRACT FROM AUTHOR]

One of the main problems of the research area of network coding is to compute good lower and upper bounds of the achievable cardinality of so-called subspace codes in \mathcal Pq(n) , i.e., the set of subspaces of \mathbb Fq^{n} , for a given minimal distance. Here we generalize a construction of Etzion and Silberstein to a wide range of parameters. This construction, named coset construction, improves or attains several of the previously best-known subspace code sizes and attains the maximum-rank distance bound for an infinite family of parameters. [ABSTRACT FROM AUTHOR]