A conversion algorithm for logarithms on GF(2n)

We present a method for expressing a root of one irreducible polynomial of degree n over GF(2) in terms of a basis of GF(2ⁿ) over GF(2) associated with another. This allows us, when both polynomials are primitive, to find logarithms relative to one polynomial from logarithms relative to the other.     

Convex subsets of 2n and bounded truth-table reducibility

Let 2ⁿ be the set of n-tuples of 0's and 1's, partially ordered componentwise. A characterization is given of the possible decompositions of arbitrary subsets of 2ⁿ as disjoint unions of sets which are convex in this ordering; this result is used to obtain a decomposition theorem for Boolean functions in terms of monotone functions. The second half of the paper contains applications to recursion theory; in particular, canonical forms for certain minimum-norm bounded-truth-table reductions are obtained.     

Subgroups of LF(2, 2n)

J. Gierster [Math. Ann. 26, 309–368] has proven a number of theorems giving the subgroup structure of LF(2, pⁿ), p an odd prime, and has given simple necessary and sufficient conditions for two elements of LF(2, pⁿ) to be conjugate. In this paper we obtain analogous theorems and conditions for LF(2, 2ⁿ). The methods involve solving congruences mod 2ⁿ.     

A note on the Hadamard product of matrices

It is shown that the smallest eigenvalue of the Hadamard product A × B of two positive definite Hermitian matrices is bounded from below by the smallest eigenvalue of AB^T.     

Spreads and packings for a class of ((2n + 1)(2n − 1 − 1) + 1, 2n − 1, 1)-designs

Resolvable designs, especially those that are multiply resolvable, have been of interest for many years. This paper takes a class of BIBD's found by Denniston and constructs several spreads and packings of these designs. Interesting ties with inversive geometry are exhibited and utilized along the way.     

On a series of Hadamard matrices of order 2 t and the maximal excess of Hadamard matrices of order 22t

In this paper we give a new series of Hadamard matrices of order 2 ^t . When the order is 16, Hadamard matrices obtained here belong to class II, class V or to class IV of Hall's classification [3]. By combining our matrices with the matrices belonging to class I, class II or class III obtained before, we can say that we have direct construction, namely without resorting to block designs, for all classes of Hadamard matrices of order 16.Furthermore we show that the maximal excess of Hadamard matrices of order 2^2t is 2^3t , which was proved by J. Hammer, R. Levingston and J. Seberry [4]. We believe that our matrices are inequivalent to the matrices used by the above authors. More generally, if there is an Hadamard matrix of order 4n ² with the maximal excess 8n ³, then there exist more than one inequivalent Hadamard matrices of order 2^2t n ² with the maximal excess 2^3t n ³ for anyt 2.     

The diophantine equation x3 + 3y3 = 2n

It is proved that the equation of the title has a finite number of integral solutions (x, y, n) and necessary conditions are given for (x, y, n) in order that it can be a solution (Theorem 2). It is also proved that for a given odd x₀ there is at most one integral solution (y, n), n ≥ 3, to x₀³ + 3y³ = 2ⁿ and for a given odd y₀ there is at most one integral solution (x, n), n ≥ 3, to x³ + 3y₀³ = 2ⁿ.     

Hadamard matrices of order 764 exist

Two Hadamard matrices of order 764 of Goethals-Seidel type are constructed. The author was supported by an NSERC Discovery Grant.     

Hadamard matrices of order 4(2p + 1)

It is shown in this paper that if p is a prime and q = 2p ? 1 is a prime power, then there exists an Hadamard matrix of order 4(2p + 1).     

The complete solution in integers of x3 + 3y3 = 2n

Some general remarks are made concerning the equation f(x, y) = qⁿ in the integral unknowns x, y, n, where f is an integral form and q > 1 is a given integer. It is proved that the only integral triads (x, y, n) satisfying x³ + 3y³ = 2ⁿ are (x, y, n) = (?1, 1, 1), (1, 1, 2), (?7, 5, 5,), (5, 1, 7).     

Three-parameter complex Hadamard matrices of order 6

A three-parameter family of complex Hadamard matrices of order 6 is presented. It significantly extends the set of closed form complex Hadamard matrices of this order, and in particular contains all previously described one- and two-parameter families as subfamilies.     

On the classification of Hadamard matrices of order 32

All equivalence classes of Hadamard matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of Hadamard matrices. We find all equivalence classes of Hadamard matrices of order 32 which are of certain types. It turns out that there are exactly 13, 680, 757 Hadamard matrices of one type and 26, 369 such matrices of another type. Based on experience with the classification of Hadamard matrices of smaller order, it is expected that the number of the remaining two types of these matrices, relative to the total number of Hadamard matrices of order 32, to be insignificant. © 2009 Wiley Periodicals, Inc. J Combin Designs 18:328–336, 2010     

Hadamard matrices of order 28 with automorphisms of order 7

The only primes which can divide the order of the automorphism group of a Hadamard matrix of order 28 are 13, 7, 3, and 2, and there are precisely four inequivalent matrices with automorphisms of order 13 (Tonchev, J. Combin. Theory Ser. A35 (1983), 43–57). In this paper we show that there are exactly twelve inequivalent Hadamard matrices of order 28 with automorphisms of order 7. In particular, there are precisely seven matrices with transitive automorphism groups.     

On the set J(v) for Steiner Quadruple systems of order v = 2n with n⩾4

In the area of the Block-Intersection problem for Steiner Quadruple Systems (see [4, 5]), we prove that q₁₆?37 = 103 and q₁₆?29 = 111 ?J (16), and that q_v?h?J(v) for h = 21, 25, v = 2ⁿ and n?4.     

A note on infinite systems of linear inequalities in Rn

A necessary and sufficient condition for infinite systems of linear inequalities to have solutions is given, using the Kuhn-Fourier Theorem, which deals with finite systems.