Construction of linear codes with large minimum distance

A natural goal in coding theory is to find a linear [n, k;q]-code such that the minimum distance d is maximal. In this paper, we introduce an algorithm to construct linear [n, k;q]-codes with a prescribed minimum distance d by constructing an equivalent structure, the so-called minihyper, which is a system of points in the (k - 1)-dimensional projective geometry P_k-1(q) over the finite field F_q with q elements. To construct such minihypers we first prescribe a group of automorphisms, transform the construction problem to a diophantine system of equations, and then apply a lattice-point-enumeration algorithm to solve this system of equations. Finally, we present a list of parameters of new codes that we constructed with the introduced method. For example, there is a new optimal [80, 4; 8]-code with minimum distance 68.