An Assmus--Mattson Theorem for Rank Metric Codes

Abstract

A t-(n, d, λ) design over Fq, or a subspace design, is a collection of ddimensional subspaces of F n q , called blocks, with the property that every t-dimensional subspace of F n q is contained in the same number λ of blocks. A collection of matrices in F n×m q is said to hold a t-design over Fq if the set of column spaces of its elements forms the blocks of a subspace design. We use notions of puncturing and shortening of rank metric codes and the rank-metric MacWilliams identities to establish conditions under which the words of a given rank in a linear rank metric code hold a t-design over Fq. We show that for Fqm -linear vector rank metric codes, the property of a code being MRD is equivalent to its minimal weight codewords holding trivial subspace designs and show that this characterization does not hold for Fq-linear matrix MRD code that are not linear over Fqm .2010 Mathematics Subject Classification. 11T71, 05B05, 05E20.

Weighted Subspace Designs from $q$-Polymatroids

“…If we apply this with Corollary 73, we retrieve the Assmus-Mattson theorem for F q m -[n, k, d] codes (c.f. [6]).…”

“…Example 76. Let C be the F 6 2 -[6, 3, 3] vector rank metric code of Example 53. Then both C and its dual have weight distribution [1,0,0,567,37044,142884,81648].…”

“…This theorem gives a criterion for identifying a t-design as a collection of supports of codewords of fixed weight in a linear code. Since its publication in 1969 it has seen a number of generalisations [6,13] and has been used widely to obtain new constructions of designs [8,16]. In one of these results [3], the authors define a weighted t-design as a generalisation of a classical t-design and give criteria for identifying such an object among the dependent sets of a matroid of a fixed cardinality.…”

On $α$-points of $q$-analogs of the Fano plane

“…Subspace designs are interlinked to the theory of network coding in various ways. To this effect we mention the recently found q-analog of the theorem of Assmus and Mattson [9], and that a t-(v, k, 1) q Steiner systems provides a (v, 2(k−t+1); k) q constant dimension network code of maximum possible size.…”

Hermitian Rank Metric Codes and Duality

“…Rank metric codes were introduced by Delsarte (1978) as a q-analogue of coding theory [13]. Due to their applications in cryptography and in network error correction ( [30] [31]), there is a great interest in studying their general properties and their connections with other topics [1], [4], [9]- [11], [24], [26], [27], [29].…”