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Abstract: In two recent papers we overhauled the theory of ternary complementary pairs, focusing on questions relating to the possible weights of pairs, and special pairs from which all others can be derived, which we call “primitive.” Of particular interest at this time is a new refinement of the concept of primitivity, which necessitates some revisions to our tables. In this article we report on the state of the art with respect to primitive pairs and elaborate on some conjectures in light of new data. 30 new primitive pairs are given; the status of 12 previously “primitive” pairs is changed to “imprimitive.” [Copyright &y& Elsevier]

Abstract: In [R. Craigen, C. Koukouvinos, A theory of ternary complementary pairs, J. Combin. Theory Ser. A 96 (2001) 358–375], we proposed a systematic approach to the theory of ternary complementary pairs (TCPs) and showed how all pairs known then could be constructed using a single elementary product, the natural equivalence relations, and a handful of pairs which we called primitive. We also introduced more new primitive pairs than could be inferred previously, concluding with some conjectures reflecting the patterns that were beginning to arise in light of the new approach. In this paper we take what appears to be the natural next step, by investigating these patterns among those lengths and weights that are within easy computational distance from the last length considered therein, length 14. We give complete results up to length 21, and partial results up to length 28. (Ironically, although we proceed analytically by weight first then length, for computational reasons we are bound, in this empirical investigation, to proceed according to length first.) Thus we provide support for the previous conjectures, and shed enough new light to speculate further as to the likely ultimate shape of the theory. Since short term work on TCPs will require massive acquisition of data about small pairs, we also discuss affixes—a computational strategy that arose out of the investigations culminating in this article. [Copyright &y& Elsevier]