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Further explorations into ternary complementary pairs. (English) Zbl 1095.05007

Summary: In [R. Craigen and C. Koukouvinos, J. Comb. Theory, Ser. A 96, 358–375 (2001; Zbl 1012.94010)], we proposed a systematic approach to the theory of ternary complementary pairs 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 ternary complementary pairs 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.

MSC:

05B30 Other designs, configurations
94A55 Shift register sequences and sequences over finite alphabets in information and communication theory

Citations:

Zbl 1012.94010
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References:

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