Abstract
Letq≡ 3 (mod 4) be a prime power and put\(n = \frac{{q  1}}{2}\). We consider a cyclic relative difference set with parametersq ^{2}−1,q, 1,q−1 associated with the quadratic extension GF(q^{2})/GF((q). The even part and the odd part of the cyclic relative difference set taken modulon are\(2  \left\{ {n;\frac{{n + 1}}{2};\frac{{n + 1}}{2}} \right\}\) supplementary difference sets. Moreover it turns out that their complementary subsets are identical with the Szekeres difference sets. This result clarifies the true nature of the Szekeres difference sets. We prove these results by using the theory of the relative Gauss sums.
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Abbreviations
 q :

a power of a primep
 F=GF (q):

a finite field withq elements
 K=GF (q ^{t}):

an extension ofF of degreet≧2
 ξ :

a primitive element ofK
 g :

a primitive element ofF
 K ^{*} :

the multiplicative group ofK
 F ^{*} :

the multiplicative group ofF
 S _{ K } :

trace fromK
 S _{ F } :

trace fromF
 S _{ K/F } :

relative trace fromK toF
 N _{ K/F } :

relative norm fromK toF
 Z:

the rational integer ring
 J _{ m }(x):

1+x+x ^{2}+...+x ^{m} −1
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 [4]
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 [5]
K.Yamamoto, On congruences arising from relative Gauss sums, in:Number Theory and Combinatorics Japan 1984, World Scientific Publ., 1985, 423–446.
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Yamada, M. On a relation between a cyclic relative difference set associated with the quadratic extensions of a finite field and the Szekeres difference sets. Combinatorica 8, 207–216 (1988). https://doi.org/10.1007/BF02122802
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AMS subject classification
 05 B 10