Circulants and difference sets

Author:
Morris Newman

Journal:
Proc. Amer. Math. Soc. **88** (1983), 184-188

MSC:
Primary 05B10; Secondary 05B20, 12C15

DOI:
https://doi.org/10.1090/S0002-9939-1983-0691306-X

MathSciNet review:
691306

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be any field, a polynomial over of degree , the full cycle, and the circulant . Assume that if is of finite characteristic . then . It is shown that the rank of over is , where is the degree of the greatest common divisor of and . This result is used to determine the rank modulo a prime of the incidence matrix associated with a difference set. The notion of the *degree* of a difference set is introduced. Certain theorems connected with this notion are proved, and an open problem is stated. Some numerical results are appended.

**[1]**M. Hall, Jr.,*A survey of difference sets*, Proc. Amer. Math. Soc.**7**(1956), 975-986. MR**0082502 (18:560h)****[2]**M. Newman,*Invariant factors of combinatorial matrices*, Israel J. Math.**10**(1971), 126-130. MR**0369103 (51:5339)****[3]**H. J. Ryser,*Combinatorial mathematics*, Carus Math. Monographs, no. 14, Math. Assoc. Amer., distributed by Wiley, New York, 1963. MR**0150048 (27:51)**

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DOI:
https://doi.org/10.1090/S0002-9939-1983-0691306-X

Article copyright:
© Copyright 1983
American Mathematical Society