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

Functional codes arising from quadric intersections with Hermitian varieties

AbstractWe investigate the functional code Ch(X) introduced by G. Lachaud (1996) [10] in the special case where X is a non-singular Hermitian variety in PG(N,q2) and h=2. In [4], F.A.B. Edoukou (2007) solved the conjecture of Sørensen (1991) [11] on the minimum distance of this code for a Hermitian variety X in PG(3,q2). In this paper, we will answer the question about the minimum distance in general dimension N, with N<O(q2). We also prove that the small weight codewords correspond to the intersection of X with the union of 2 hyperplanes

On the small weight codewords of the functional codes C_2(Q), Q a non-singular quadric

We study the small weight codewords of the functional code C_2(Q), with Q a non-singular quadric of PG(N,q). We prove that the small weight codewords correspond to the intersections of Q with the singular quadrics of PG(N,q) consisting of two hyperplanes. We also calculate the number of codewords having these small weights

Subspace code constructions

We improve on the lower bound of the maximum number of planes of ${\rm PG}(8,q)$ mutually intersecting in at most one point leading to the following lower bound: ${\cal A}_q(9, 4; 3) \ge q^{12}+2q^8+2q^7+q^6+q^5+q^4+1$ for constant dimension subspace codes. We also construct two new non-equivalent $(6, (q^3-1)(q^2+q+1), 4; 3)_q$ constant dimension subspace orbit-codes