Construction of linear codes with large minimum distance

A natural goal in coding theory is to find a linear In, k; q]-code such that the minimum distance d is maximal. In this paper, we introduce an algorithm to construct linear In, 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 - I)-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 [8 0, 4; 81 -code with minimum distance 68.