The T4 and G4 constructions for Costas arrays

B.Ya. Ryabko, " Data compression by means of a book stack, Probl. Abstract-Two of the algebraic constructions for Costas arrays, designated as T4 and C,, are described in detail, and necessary and sufficient conditions are given for the sizes of Costas arrays for which these constructions occur. These constructions depend on the existence of primitive roots satisfying certain equations in finite fields. In [l], a number of systematic algebraic constructions for Costas Arrays are described. The validity of several of these constructions is proved in [2]. However, two of the algebraic constructions, designated T, and G4 in [l], are not discussed in [2]. The present note proves the assertions made in [I] concerning these two constructions , and furnishes some additional algebraic information. We briefly review a few of the definitions from 111 and 121. Definition I : A Costas array of order n is an n x n permutation matrix with the property that the vectors connecting two 1's of the matrix are all distinct as vectors. (That is, no two vectors are equal in both magnitude and slope).Definition 2: If n = q-2, where q = pk is the size of a finite field, then the Lempel construction L , for a Costas array of order n sets a,, = 1 iff a' + aJ = 1, 1 i i , j 5 q-2, where a is any fixed primitive root in GF(q). (Note that the Lempel construction gives a symmetric permutation matrix with the Costas property.) Definition 3: The T4 construction occurs for GF(q) iff there is a primitive element a in GF(q) with a + a2 = 1. the primitive root a which satisfies a + a' = 1 in GF(q), we have both a' + a2 = 1 and a' + a' = 1. Thus both a,' = 1 and a,, = 1 in the L , construction of order q-2. Removing the two topmost rows and the two leftmost columns from the L , array leaves a Costas array of order q-4, which is the Theorem I : A necessary condition for the T, construction is that q is 4, or 5, or 9, or a prime p with p = k 1 (mod 10). Proof: We are asking for a field GF(q) in which the equation x2 + x-1 has roots, and in which at least one of these roots is primitive in …