We investigate subspace codes whose codewords are subspaces of PG(4, q) having nonconstant dimension. In particular, examples of optimal mixed-dimension subspace codes are provided, showing that A q (5, 3) = 2(q 3 + 1).
IntroductionLet V be an n-dimensional vector space over GF(q), q any prime power. The set S(V ) of all subspaces of V , or subspaces of the projective space PG(V ), forms a metric space with respect to the subspace distance defined by d s (U, U ′ ) = dim(U + U ′ ) − dim(U ∩ U ′ ). In the context of subspace codes, the main problem is to determine the largest possible size of codes in the space (S(V ), d s ) with a given minimum distance, and to classify the corresponding optimal codes. The interest in these codes is a consequence of the fact that codes in the projective space and codes in the Grassmannian over a finite field referred to as subspace codes and constantdimension codes (CDC), respectively, have been proposed for error control in random linear network coding. An (n, M, d) q mixed-dimension subspace code is a set C of subspaces of V with |C| = M and minimum subspace distanceThe maximum size of an (n, M, d) q mixed-dimension subspace code is denoted by A q (n, d). In this paper we discuss the smallest open mixed-dimension case, which occurs when n = 5 and d = 3. In particular, examples of optimal mixed-dimension subspace codes are provided, showing that A q (5, 3) = 2(q 3 + 1).We wish to remark that, independently, also Honold, Kiermaier and Kurz proved that A q (5, 3) = 2(q 3 + 1). We refer to their article [5] for their construction method, and for many other results on subspace codes.
