Partial covers of PG(n,q)

AbstractIn this paper, we show that a set of q+a hyperplanes, q>13, a≤(q−10)/4, that does not cover PG(n,q), does not cover at least qn−1−aqn−2 points, and show that this lower bound is sharp. If the number of non-covered points is at most qn−1, then we show that all non-covered points are contained in one hyperplane. Finally, using a recent result of Blokhuis, Brouwer and Szőnyi [8], we remark that the bound on a for which these results are valid can be improved to a<(q−2)/3 and that this upper bound on a is sharp

Binary and Ternary Quasi-perfect Codes with Small Dimensions

The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First we give a list of infinite families of QP codes which includes all binary, ternary and quaternary codes known to is. We continue further with a list of sporadic examples of binary and ternary QP codes. Later we present the results of our investigation where binary QP codes of dimensions up to 14 and ternary QP codes of dimensions up to 13 are classified.Comment: 4 page

An improvement of the Griesmer bound for some small minimum distances

AbstractIn this paper we give some lower and upper bounds for the smallest length n(k, d) of a binary linear code with dimension k and minimum distance d. The lower bounds improve the known ones for small d. In the last section we summarize what we know about n(8, d)

Asymptotic Bound on Binary Self-Orthogonal Codes

We present two constructions for binary self-orthogonal codes. It turns out that our constructions yield a constructive bound on binary self-orthogonal codes. In particular, when the information rate R=1/2, by our constructive lower bound, the relative minimum distance \delta\approx 0.0595 (for GV bound, \delta\approx 0.110). Moreover, we have proved that the binary self-orthogonal codes asymptotically achieve the Gilbert-Varshamov bound.Comment: 4 pages 1 figur

Some constructions of superimposed codes in Euclidean spaces

AbstractWe describe three new methods for obtaining superimposed codes in Euclidean spaces. With help of them we construct codes with parameters improving upon known constructions. We also prove that the spherical simplex code is not optimal as superimposed code at least for dimensions greater than 9

Partial spreads and vector space partitions

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.Comment: 30 pages, 1 tabl