Minimum distance of Symplectic Grassmann codes

We introduce the Symplectic Grassmann codes as projective codes defined by symplectic Grassmannians, in analogy with the orthogonal Grassmann codes introduced in [4]. Note that the Lagrangian-Grassmannian codes are a special class of Symplectic Grassmann codes. We describe the weight enumerator of the Lagrangian--Grassmannian codes of rank $2$ and $3$ and we determine the minimum distance of the line Symplectic Grassmann codes.Comment: Revised contents and biblograph

Line Polar Grassmann Codes of Orthogonal Type

Polar Grassmann codes of orthogonal type have been introduced in I. Cardinali and L. Giuzzi, \emph{Codes and caps from orthogonal Grassmannians}, {Finite Fields Appl.} {\bf 24} (2013), 148-169. They are subcodes of the Grassmann code arising from the projective system defined by the Pl\"ucker embedding of a polar Grassmannian of orthogonal type. In the present paper we fully determine the minimum distance of line polar Grassmann Codes of orthogonal type for $q$ odd

On the equivalence of linear sets

Let $L$ be a linear set of pseudoregulus type in a line $\ell$ in $\Sigma^*=\mathrm{PG}(t-1,q^t)$, $t=5$ or $t>6$. We provide examples of $q$-order canonical subgeometries $\Sigma_1,\, \Sigma_2 \subset \Sigma^*$ such that there is a $(t-3)$-space $\Gamma \subset \Sigma^*\setminus (\Sigma_1 \cup \Sigma_2 \cup \ell)$ with the property that for $i=1,2$, $L$ is the projection of $\Sigma_i$ from center $\Gamma$ and there exists no collineation $\phi$ of $\Sigma^*$ such that $\Gamma^{\phi}=\Gamma$ and $\Sigma_1^{\phi}=\Sigma_2$. Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des. Codes Cryptogr. 56:89-104, 2010) states the existence of a collineation between the projecting configurations (each of them consisting of a center and a subgeometry), which give rise by means of projections to two linear sets. It follows from our examples that this condition is not necessary for the equivalence of two linear sets as stated there. We characterize the linear sets for which the condition above is actually necessary.Comment: Preprint version. Referees' suggestions and corrections implemented. The final version is to appear in Designs, Codes and Cryptograph

Line Hermitian Grassmann codes and their parameters

In this paper we introduce and study line Hermitian Grassmann codes as those subcodes of the Grassmann codes associated to the 2-Grassmannian of a Hermitian polar space defined over a finite field. In particular, we determine the parameters and characterize the words of minimum weight

Nonintersecting Subspaces Based on Finite Alphabets

Two subspaces of a vector space are here called ``nonintersecting'' if they meet only in the zero vector. The following problem arises in the design of noncoherent multiple-antenna communications systems. How many pairwise nonintersecting M_t-dimensional subspaces of an m-dimensional vector space V over a field F can be found, if the generator matrices for the subspaces may contain only symbols from a given finite alphabet A subseteq F? The most important case is when F is the field of complex numbers C; then M_t is the number of antennas. If A = F = GF(q) it is shown that the number of nonintersecting subspaces is at most (q^m-1)/(q^{M_t}-1), and that this bound can be attained if and only if m is divisible by M_t. Furthermore these subspaces remain nonintersecting when ``lifted'' to the complex field. Thus the finite field case is essentially completely solved. In the case when F = C only the case M_t=2 is considered. It is shown that if A is a PSK-configuration, consisting of the 2^r complex roots of unity, the number of nonintersecting planes is at least 2^{r(m-2)} and at most 2^{r(m-1)-1} (the lower bound may in fact be the best that can be achieved).Comment: 14 page