Construction of Optimal Linear Codes Using Flats and Spreads in a Finite Projective Geometry

In this paper, we shall consider a problem of constructing an optimal linear code whose code length n is minimum among (*, k, d ; s)-codes for given integers k, d and s. In [5], we showed that this problem is equivalent to Problem B of a linear programming which has some geometrical structure and gave a geometrical method of constructing a solution of Problem B using a set of flats in a finite projective geometry and obtained a necessary and sufficient conditions for integers k, d and s that there exists such a geometrical solution of Problem B for given integers k, d and s. But there was no space to give the proof of the main theorem 4.2 in [5]. The purpose of this paper is to give the proof of [5, Theorem 4.2], i.e. to give a systematic method of constructing a solution of Problem B using flats and spreads in a finite projective geometry

Optimal Binary Subspace Codes of Length 6, Constant Dimension 3 and Minimum Distance 4

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 $\operatorname{PG}(5,2)$ 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

Classification of large partial plane spreads in PG(6,2)PG(6,2) and related combinatorial objects

In this article, the partial plane spreads in $PG(6,2)$ of maximum possible size $17$ and of size $16$ are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: Vector space partitions of $PG(6,2)$ of type $(3^{16} 4^1)$, binary $3\times 4$ MRD codes of minimum rank distance $3$, and subspace codes with parameters $(7,17,6)_2$ and $(7,34,5)_2$.Comment: 31 pages, 9 table