Ninth International Workshop on Algebraic and Combinatorial Coding Theory 19-25 June, 2004, Kranevo, Bulgaria pp. 101–106 One-generator quasi-cyclic ternary linear codes 1 Rumen Daskalov daskalov@tugab.bg Department of Mathematics, Technical University of Gabrovo, 5300 Gabrovo, BULGARIA Plamen Hristov plhristov@tugab.bg Department of Mathematics, Technical University of Gabrovo, 5300 Gabrovo, BULGARIA Abstract. One of the main problems in coding theory is to construct codes with best possible minimum distances. In this paper, twenty new ternary codes are constructed, which improve the best known lower bounds on minimum distance. 1 Introduction Let GF (q) denote the Galois field of q elements. A linear code C over GF (q) of length n, dimension k and minimum Hamming distance d is called an [n, k, d] q code. A code C is said to be quasi-cyclic (QC or p-QC) if a cyclic shift of a codeword by p positions results in another codeword. A cyclic shift of an m- tuple (x 0 ,x 1 ,...,x m−1 ) is the m-tuple (x m−1 ,x 0 ,...,x m−2 ). The blocklength, n, of a p-QC code is a multiple of p, so that n = pm [3]. A matrix B of the form B =         b 0 b 1 b 2 ··· b m−2 b m−1 b m−1 b 0 b 1 ··· b m−3 b m−2 b m−2 b m−1 b 0 ··· b m−4 b m−3 . . . . . . . . . . . . . . . b 1 b 2 b 3 ··· b m−1 b 0         , (1) is called a circulant matrix. A class of QC codes can be constructed from m × m circulant matrices (with a suitable permutation of coordinates [8]). In this case, the generator matrix, G, can be represented as G =[B 1 ,B 2 , ..., B p ] , (2) 1 This work was partially supported by the Bulgarian National Science Fund under Contract in TU–Gabrovo.