Let q be an odd prime power. In [2], van Lint and MacWilliams conjectured that the only q-subset X if GF(q^2), with the properties 0,1 belong X and x-y is a square for all x,y from X, is the set GF(q). For q a prime this is a consequence (due to van Lint and MacWilliams) of a theorem of Rédei, [4, p. 237 Satz 24']. This case was also proved in an elementary way by Lovász and Schrijver [3]. The problem arose in an attempt to characterise the vectors of minimum weight in certain quadratic residue-codes. In [1], Blokhuis confirmed the conjecture.
In this lecture, we discuss the Blokhuis' proof in detail.
References:
[1] A. Blokhuis, On subsets of GF(q^2) with square differences, Indag. Math. 46 (1984) 369-372.
[2] J. H. van Lint, F. J. MacWilliams, Generalized Quadratic Residue Codes, IEEE Transactions on Information Theory, IT 24, 730--737 (1978).
[3] L. Lovász, A. Schrijver, Remarks on a theorem of Rédei, Studia Scientiarum Mathematicarum Hungarica 16, 449--454 (1981).
[4] L. Rédei, Lűckenhafte Polynome űber endlichen Kőrpern, Birkhäuser Verlag, Base und Stuttgart, 1970.
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