The triple distribution of codes and ordered codes

We study the distribution of triples of codewords of codes and ordered codes. Schrijver [A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory 51 (8) (2005) 2859-2866] used the triple distribution of a code to establish a bound on the number of codewords based on semidefinite programming. In the first part of this work, we generalize this approach for ordered codes. In the second part, we consider linear codes and linear ordered codes and present a MacWilliams-type identity for the triple distribution of their dual code. Based on the non-negativity of this linear transform, we establish a linear programming bound and conclude with a table of parameters for which this bound yields better results than the standard linear programming bound.

► We study the distribution of triples of codewords of codes and ordered codes. ► In the first part, we generalize Schrijver's SDP bound to the case of ordered codes. ► In the second part, we present a MacWilliams-type identity for the triple distribution. ► Based on the MacWilliams transform for triples, we establish a new LP bound. ► We give code parameters for which the new bound is better than the standard LP bound.