Abstract. Generalized quasi-cyclic (GQC) codes form a natural generalization of quasi-cyclic (QC) codes. They are viewed here as mixed alphabet codes over a family of ring alphabets. Decomposing these rings into local rings by the Chinese Remainder Theorem yields a decomposition of GQC codes into a sum of concatenated codes. This decomposition leads to a trace formula, a minimum distance bound, and to a criteria for the GQC code to be self-dual or to be linear complementary dual (LCD). Explicit long GQC codes that are LCD, but not QC, are exhibited.

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A perfect matroid design (PMD) is a matroid whose flats of the same rank all have the same size. In this paper we introduce the q-analogue of a PMD and its properties. In order to do that, we first establish new cryptomorphic definitions for q-matroids. We show that q-Steiner systems are examples of q-PMD's and we use this matroid structure to construct subspace designs from q-Steiner systems. We apply this construction to S(2, 13, 3; q) Steiner systems and hence establish the existence of subspace designs with previously unknown parameters.

A perfect matroid design (PMD) is a matroid whose flats of the same rank all have the same size. In this paper we introduce the q-analogue of a PMD and its properties. In order to do so, we first establish a new cryptomorphic definition for q-matroids. We show that q-Steiner systems are examples of q-PMD’s and we use this q-matroid structure to construct subspace designs from q-Steiner systems. We apply this construction to the only known q-Steiner system, which has parameters S(2, 3, 13; 2), and hence establish the existence of a new subspace design with parameters 2-(13, 4, 5115; 2).

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