Recursive MDS-Codes and Pseudogeometries

Abstract

In [2,4] the notion of a recursive code was introduced and some constructions of recursive MDS codes were proposed. The main result was that for any q ∉ {2, 6} (except possibly ∈ {14, 18, 26, 42}) there exists a recursive MDS-code in an alphabet of q elements of length 4 and combinatorial dimension 2(i.e. a recursive {4, 2, 3}_q-code). One of the constructions we used there was that of pseudogeometries; it enabled us to show that for any q > 126 (except possibly q = 164) there exists a recursive [4,2,3]_q-code that contains all the “constants”. One part of the present note is the further application of the pseudogeometry construction which shows that for any q > 164 (resp. q > 26644) there exists a recursive [7,2,6]_q-code (resp. [13,2,12]_q-code) containing “constants”. Another result presented here is a negative one: we show that there is no nontrivial pseudogeometry consisting of 14, 18, 26 or 42 points with no lines of order 2, 3, 4 or 6, so the pseudogeometry construction cannot be applied for settling the question mentioned in the above. In both cases the usage of computer is essential.