Arch. Math., Vol. 62, 126-133 (1994) 0003-889X/94/6202-0126 $ 3.10/0 9 1994 Birkh/iuser Verlag, Basel A note on orthogonal circulant matrices over finite fields By DIETER JUNGNICKEL, THOMAS BETH and WILLI GEISELMANN Introduction. Orthogonal circulant matrices over finite fields are of interest in Cod- ing Theory, cf. [8]. They can also be used to generate all self-dual normal bases of GF(q")/GF(q) (see [6] for general background on finite fields and [3] for a detailed treatment of normal and self-dual basesl provided that one such basis is known and, in particular, to determine the number of such bases; see [1] and [4]. To this purpose, one requires the order of the group OC (n, q) of orthogonal circulant tn x n)-matrices over GF (q). This problem was solved by MacWilliams [7] in the special case where q is a prime. In [2] Byrd and Vaughan give a formula for arbitrary prime powers q. The problem of determining IOC (n, q) for arbitrary q was also considered by Beth and Geiselmann [1] because of the connection to self-dual normal bases mentioned above. Accordingly, these authors dealt explicitly only with the cases where either q is even and n not divisible by 4, or both q and n odd. (By a result of Lempel & Weinberger [6], these are the conditions for the existence of a self-dual normal basis). They give a simple algorithm to construct the orthogonal circulant (n x n)-matrices, but as we shall see, the proof of [1] is only partially correct. In the present note, we shall give a much simpler proof for the general case than the one given in [2]. Furthermore, our construction of the orthogonal circulant In x n)-ma- trices is simpler and does not need the factorization of x" 1 over GF (q). The most interesting case in our proof is when both q and n are even. In fact the proof of MacWilliams carries over to the general case without any problems in all other cases. Therefore we shall treal only the case q and n even in detail. A detail proof for all cases can be found in [3]. In [1], [2] and [7], the obvious isomorphism between the algebra of circulant (n x n)-ma- trices over GF (q) and the algebra Rn, q = GF (q)[x]/(x" - 1) is used. The following basic observation characterizes those polynomials in R,,~ that correspond to orthogonal ma- trices in C (n, q). Lemma. A circulant (n x n)-matrix C over GF(q) with first row (c o, c I .... , c~-1) is orthogonal if and only if the corresponding polynomial c(x)= c o + clx + ""+ c._lx"-16R.,~ satisfies c(x)cr(x) - 1, where cr(x):= co + c.-l x + '" + c2 Xn-2 + c~ x"- ~ is the polynomial corresponding to the transpose C r of C. N o t a t i o n. The polynomial c r (x) is called the transpose of c (x) and a polynomial in R.,q satisfiying c (x)c r (x) = t an orthogonal polynomial, whereas a polynomial with