Update on the Extension of Good Linear Codes

In this short note we state how we construct new good linear codes C over the finite field with q elements. We start with already good (= high minimum distance d for given length n and dimension k) codes which we got for example by our method [M. Braun. Construction of linear codes with large minimum distance, IEEE Transactions on Information Theory, 50(8):1687–1691, 2004; M. Braun, A. Kohnert, and A. Wassermann. Construction of ( n , r ) -arcs in PG(2, q), Innovations in Incidence Geometry, 1:133–141, 2005; M. Braun, A. Kohnert, and A. Wassermann. Construction of (sometimes) optimal linear codes, Bayreuther Mathematische Schriften, 74:69–75, 2005; M. Braun, A. Kohnert, and A. Wassermann. Optimal linear codes from matrix groups, IEEE Transactions on Information Theory, 12:4247–4251, 2005]. The advantage of this method is that we explictly get the words of minimum weight d. We try to extend the generator matrix of C by adding columns with the property that at least s of the letters added to the codewords are different from 0. Using this we know that the minimum distance of the extended code is d + s as long as the second smallest weight was ⩾ d + s . s note we only state the method and the results. A full version [Axel Kohnert. Extension of good linear codes, submitted, 6 pages, 2006] is submitted to the proceedings of Combinatorics 2006.