Projective two-weight codes with small parameters and their corresponding graphs

Let GF(q) be the n-dimensional vector space over the Galois field GF(q). The Hamming distance between two vectors of GF(q) is defined to be the number of coordinates in which they differ. A q-ary linear [n, k, d; q]-code is a k-dimensional linear subspace of GF(q) with minimum distance d. Let n(k, d) denote the smallest value of n for which an [n, k, d]-code exists. An [n(k, d), k, d]-code is called optimal. A generator matrix G of a linear [n, k; q]-code C is any matrix of rank k over GF(q) with rows from C. Let C1 and C2 be two linear [n, k; q]-codes. They are said to be equivalent if the codewords of C2 can be obtained from the codewords of C1 via a finite sequence of transformations of the following types: (1) permutation on the set of coordinate positions; (2) multiplication of the elements in a given position by a non-zero element of GF(q); (3) application of a field automorphism to the elements in all coordinate positions. An automorphism of a linear code C is a sequence of transformations of type (1)-(3) which maps each codeword of C onto a codeword of C. All the automorphisms of a code C form a group, which is called the automorphism group Aut(C) of the code C. A linear code is called projective if no two columns of the generator matrix are linearly dependent. The weight w(x) of a codeword x is defined as the number of the non-zero entries of x. The weight enumerator of C is WC(y) = ∑n i=0 Aiy i where Ai is Corresponding author. Department of Applied Mathematics and Computer Science, Ghent University, Belgium. E-mail: [email protected] Partially supported by the Bulgarian National Science Fund under Contract MM1304/2003. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences. E-mail: [email protected] Institut fr Algebra und Geometrie, Otto-von-Guericke-Universitt, Magdeburg, Germany. E-mail: [email protected] Department of Applied Mathematics and Computer Science, Ghent University, Belgium. E-mail: [email protected]