Cyclotomic conditions leading to new Steiner 2-designs
Sometimes the validity of elementary cyclotomic conditions in a finite field is sufficient to guarantee the existence of some Steiner 2-designs with a nice group of automorphisms. For instance, R. C. Bose proved that if \(q= 12t+ 1\) (resp. \(q= 20t+1\)) is a prime power such that \(-3\) (resp. 5) is not a fourth power in \(\text{GF}(q)\), then there exists a point-regular \(S(2,4,q)\) (resp. \(S(2,5,q)\)). Cyclotomic conditions weaker than those of R. C. Bose may be found in [\textit{M. Buratti}, J. Comb. Des. 3, No. 1, 15-24 (1995; Zbl 0833.05009)]. In the present paper, the author obtained new cyclotomic conditions starting from a paper by \textit{S. Bagchi} and \textit{B. Bagchi} [J. Comb. Theory, Ser. A 52, No. 1, 51-61 (1989; Zbl 0679.05008)], where they give constructions for Steiner 2-designs generated by a relative difference family \(\mathbb{D}\) over the additive group of a Galois ring. He pointed out that the construction of S. Bagchi and B. Bagchi were previously considered by \textit{R. Mathon} [Ann. Discrete Math. 34, 353-362 (1987; Zbl 0647.05015), Theorem 3], but, unlike them, he did not give explicit cyclotomic conditions which are a consequence of the constructions. The author also pointed out that similar constructions for relative difference families were recently given by M. Greig. The method of M. Greig is very successful but needs the aid of a computer and yields difference families that have multipliers of orders smaller than those of difference families given here. The results below are the main consequences of the constructions given by the author. (i) Let \(q= 6t+1\) be a prime power and let \(3^e\) be the largest power of 3 dividing \(t\). Then, if 3 is a \(3^e\)th power but not a \(3^{e+1}\)th power in \(\text{GF}(q)\), there exists a point-regular \(S(2,7,7q)\). (ii) Let \(q= 8t+1\) be a prime power and let \(2^e\) be the largest power of 2 dividing \(t\). Then, if 2 is a \(2^{e+2}\)th power while \(1+\sqrt 2\) is a \(2^e\)th power but not a \(2^{e+1}\)th power in \(\text{GF}(q)\), there exists a point-regular \(S(2,9,9q)\). (iii) Let \(q= 10t+1\) be a prime power and let \(5^e\) be the largest power of 5 dividing \(t\). Then, if \(w\) is a primitive fifth root of unity in \(\text{GF}(q)\) such that both \(w^2(w-1)\) and \(w^4(w+1)\) are \(5^{e+1}\)th powers in \(\text{GF}(q)\), there exists a point-regular \(S(2,11,11q)\). (iv) Let \(q= 12t+1\) be a prime power and let \(3^e\) be the largest power of 3 dividing \(t\). Then, if 3 and \(2+\sqrt 3\) are \(3^e\)th powers but not \(3^{e+1}\)th powers in \(\text{GF}(q)\) and if 6 is not a \(3^{e+1}\)th power, there exists a point-regular \(S(2,13,13q)\). The results (i), (ii), (iii) improve the results by S. Bagchi and B. Bagchi (corresponding the cases where \(e=0\)) while (iv) is completely new. A less nice condition leading to point-regular \(S(2,17,17q)\) is also reported. Thus the author has improved the construction of S. Bagchi and B. Bagchi for Steiner 2-designs which are point-regular over the additive group of a Galois ring, and, as a consequence, got new cyclotomic conditions leading to point-regular Steiner 2-designs with block size \(k\in \{7,9, 11,13,17\}\).
scientific article; zbMATH DE number 1199964