A (v, k, λ) difference set D is a set of k distinct residues {a1, a2, ... , ak } modulo v such that every residue b ≢ 0 (mod v) can be expressed in exactly λ ways in the form b ≡ ai — aj (mod v). With each difference set we may associate a binary periodic sequence (s1, s2 , ...) with si = 1 if i (mod v) is in D, and si = 0 otherwise. Since this sequence is periodic of period v, we need only consider one cycle from the sequence. Such cycles we agree to call (binary) difference cycles. Difference cycles (equivalently, difference sets) have been studied intensively (2, 4). They have important applications to digital communications, mainly because they have 2-level autocorrelation. In this paper we shall point out certain other (equivalent) properties of difference cycles which seem susceptible to immediate generalization, but show that these generalizations are vacuous.
McEliece, Robert J. A Generalization of Difference Sets. Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 206-211. doi: 10.4153/CJM-1967-013-1
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author = {McEliece, Robert J.},
title = {A {Generalization} of {Difference} {Sets}},
journal = {Canadian journal of mathematics},
pages = {206--211},
year = {1967},
volume = {19},
number = {1},
doi = {10.4153/CJM-1967-013-1},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1967-013-1/}
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