A Connection Between Coding Theory and Polarized Partition Relations

means that every matrix a × b with entries in the set {0, 1, . . . , q − 1} always contains a constant i × j submatrix. The negation of (1) is denoted by (a, b) 6→ (i, j)q . The problem of deciding whether (1) holds was posed by Erdös and Rado [7] (more generally and in a different context), which also can be restated in terms of graph theory or as a variant of the celebrated Ramsey problem. Several authors, for example in [6, 11, 12, 15, 20], have considered distinct approaches of this partition, either studying related problems, or investigating connections to other combinatorial structures. In particular, we remark that Hadamard matrices and orthogonal Latin squares yield classes of polarized partitions (see [1, 4]). Since such combinatorial concepts are also used to construct error-correcting codes (see [18, 19]), it seems interesting to investigate how codes and polarized partitions are related. We are concerned with the above question. The main result (Theorem 1) is to establish a connection between polarized partition relation and a central problem in coding theory, described as follows: determine the maximum number Aq(n, d) of codewords in a q-ary code of length n and minimum distance d. This connection constitutes a systematic way of constructing polarized partitions. In particular, bounds for the extremal problem Pq(2, t) = min{n ∈ N : (n, n) → (2, t)q} are determined (Section 4), which imply previous results from [1, 4]. In contrast to the case where q = 2, our knowledge on exact values of Pq(2, t) is rather poor when q ≥ 3 and t ≥ 2. Indeed, the topic is so short of construction that the only exact value known is P3(2, 2) = 11, by Exoo [8]. However, making use of the table for A3(n, d) due to Vaessen et al. [21], we derive good bounds for P3(2, t) when t is small. Moreover, the method produces a relationship between Aq(n, d) and the classical Zarankiewicz numbers (Section 5). As a consequence, the Plotkin bound is obtained from a result due to Hyltén-Cavallius [14].