It is shown that the maximum size of a binary subspace code of packet length v = 6, minimum subspace distance d = 4, and constant dimension k = 3 is M = 77; in Finite Geometry terms, the maximum number of planes in PG(5, 2) mutually intersecting in at most a point is 77. Optimal binary (v, M, d; k) = (6, 77, 4; 3) subspace codes are classified into 5 isomorphism types, and a computer-free construction of one isomorphism type is provided. The construction uses both geometry and finite fields theory and generalizes to any q, yielding a new family of q-ary (6, q 6 + 2q 2 + 2q + 1, 4; 3) subspace codes.2000 Mathematics Subject Classification. Primary 94B05, 05B25, 51E20; Secondary 51E14, 51E22, 51E23.
ABSTRACT. Constant-dimension codes with the maximum possible minimum distance have been studied under the name of partial spreads in Finite Geometry for several decades. Not surprisingly, for this subclass typically the sharpest bounds on the maximal code size are known. The seminal works of Beutelspacher and Drake & Freeman on partial spreads date back to 1975, and 1979, respectively. From then until recently, there was almost no progress besides some computer-based constructions and classifications. It turns out that vector space partitions provide the appropriate theoretical framework and can be used to improve the long-standing bounds in quite a few cases. Here, we provide a historic account on partial spreads and an interpretation of the classical results from a modern perspective. To this end, we introduce all required methods from the theory of vector space partitions and Finite Geometry in a tutorial style. We guide the reader to the current frontiers of research in that field, including a detailed description of the recent improvements.
We determine the maximum size A 2 (8, 6; 4) of a binary subspace code of packet length v = 8, minimum subspace distance d = 6, and constant dimension k = 4 to be 257. There are two isomorphism types of optimal codes. Both of them are extended LMRD codes. In finite geometry terms, the maximum number of solids in PG(7, 2) mutually intersecting in at most a point is 257. The result was obtained by combining the classification of substructures with integer linear programming techniques. This result implies that the maximum size A 2 (8, 6) of a binary mixed-dimension subspace code of packet length 8 and minimum subspace distance 6 is 257 as well.
The aim of this paper is to develop a theory of linear codes over finite chain rings from a geometric viewpoint. Generalizing a well-known result for linear codes over fields, we prove that there exists a one-to-one correspondence between so-called fat linear codes over chain rings and multisets of points in projective Hjelmslev geometries, in the sense that semilinearly isomorphic codes correspond to equivalent multisets and vice versa. Using a selected class of multisets we show that certain MacDonald codes are linearly representable over nontrivial chain rings.
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