Note on a theorem of Zaremba

JOURNAL OF COMBINATORIALTHEORY 6, 208-209 (1969)

Note on a Theorem of Zaremba GERALD LOSEY*

Department of Mathematics, Universityof Manitoba, Winnipeg19, Manitoba Communicated by N. S. Mendelsohn Received July 12, 1968

Let S be a set of q elements and S" = S • S • ... • S the n-th Cartesian power of S. The elements o f S '~ will be called vectors. A vector ( x l , x2 ..... x,) covers (Yl, Y2 ..... y,) if and only if x~ = y~ except for possibly one value of i = 1, 2,..., n. The number of vectors covered by a given vector is v = 1 § n(q -- 1). A subset M _CS" is a covering of S n if each vector o f S" is covered by at least one vector of M. We define o(n, q) = min{l M r} where M runs through all coverings of S ". If M is a covering o f S" then we must have v ] M I > ~ l S " l = q " and so I M [ >j q"/v. Hence g(n, q) ~ q"/v. A covering M of S" is perfect if each vector of S" is covered by exactly one vector o f M. In this case I M 1 = q"/v = o(n, q). The object of this note is to give a short simple p r o o f of the ,following result of Zaremba [1]: THEOREM. f f q k a power o f a prime and v = 1 + n(q -- 1) & a power q~ of q then there exists a perfect covering M o f S" of cardinality q"-~. Hence (q~ -- 1 q) = q,,_~. ~q--l'

PROOF: Let S = GF(q) be the finite field of q elements. Then S n is a vector space of dimension n over S. Let e I , e2 ..... e. be the standard basis for S ". We assume

v = 1 +n(q--

1) = qr.

Let V be a vector space of dimension r over S. The number of 1-dimensional subspaces of Vis ( q ' - - 1 ) / ( q - l) = n. * This work was sponsored by the S u m m e r Research Institute o f the C a n a d i a n Mathematical Congress.

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NOTE ON A THEOREM OF ZAREMBA

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Let v l , v2 ,..., vn be non-zero vectors of V, one on each 1-dimensional subspace of V. Define a linear t r a n s f o r m a t i o n T: S '~ - + V by T(ei) --- vf. Let M be the kernel of T. Then, since T is onto, M has d i m e n s i o n n - - r over S a n d so I M [ = qn-~. Let x E S n. T h e n T(x) ~ Avi for some vi a n d some AES. Therefore, T(x--Ae,)---- A v i - - A v i = O a n d so x - - Ae~ ---- m ~ M. Thus x = m + Ae~ a n d x differs from m in at most the i-th coordinate. Hence M is a covering of S ~ of order qn-r = q"/v.

REFERENCE 1. S. K. ZAREMBA,Covering Problems concerning Abelian Groups, J. London Math. Soc. 27 (1952), 242-246.