A construction of binary Golay sequence pairs from odd‐length Barker sequences

Binary Golay sequence pairs exist for lengths 2, 10 and 26 and, by Turyn's product construction, for all lengths of the form 2^a10^b26^c where a, b, c are non‐negative integers. Computer search has shown that all inequivalent binary Golay sequence pairs of length less than 100 can be constructed from five “seed” pairs, of length 2, 10, 10, 20 and 26. We give the first complete explanation of the origin of the length 26 binary Golay seed pair, involving a Barker sequence of length 13 and a related Barker sequence of length 11. This is the special case m=1 of a general construction for a length 16m+10 binary Golay pair from a related pair of Barker sequences of length 8m+5 and 8m+3, for integer m≥0. In the case m=0, we obtain an alternative explanation of the origin of one of the length 10 binary Golay seed pairs. The construction cannot produce binary Golay sequence pairs for m>1, having length greater than 26, because there are no Barker sequences of odd length greater than 13. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 478–491, 2009     

A note on the nonexistence of Barker sequences

A Barker sequence is a sequence with elements ±1 such that all out-of-phase aperiodic autocorrelation coefficients are 0, 1 or -1. It is known that if a Barker sequence of length s > 13 exists then s = 4N ² for some odd integer N 55, and it has long been conjectured that no such sequence exists. We review some previous attempts to improve the bound on N which, unfortunately, contain errors. We show that a recent theorem of Eliahou et al. [5] rules out all but six values of N less than 5000, the smallest of which is 689.     

Estimates for Wieferich numbers

We define Wieferich numbers to be those odd integers w≥3 that satisfy the congruence 2 ^φ(w)≡1 (mod  w ²). It is clear that the distribution of Wieferich numbers is closely related to the distribution of Wieferich primes, and we give some quantitative forms of this statement. We establish several unconditional asymptotic results about Wieferich numbers; analogous results for the set of Wieferich primes remain out of reach. Finally, we consider several modifications of the above definition and demonstrate that our methods apply to such sets of integers as well. During the preparation of this paper, W.B. was supported in part by NSF grant DMS-0070628, F.L. was supported in part by grants SEP-CONACYT 37259-E and 37260-E, and I.S. was supported in part by ARC grant DP0211459.     

A search for Wieferich and Wilson primes

An odd prime is called a Wieferich prime if

alternatively, a Wilson prime if

To date, the only known Wieferich primes are and , while the only known Wilson primes are , and . We report that there exist no new Wieferich primes , and no new Wilson primes . It is elementary that both defining congruences above hold merely (mod ), and it is sometimes estimated on heuristic grounds that the ``probability" that is Wieferich (independently: that is Wilson) is about . We provide some statistical data relevant to occurrences of small values of the pertinent Fermat and Wilson quotients (mod ).

     

The continuing search for Wieferich primes

A prime satisfying the congruence

is called a Wieferich prime. Although the number of Wieferich primes is believed to be infinite, the only ones that have been discovered so far are and . This paper describes a search for further solutions. The search was conducted via a large scale Internet based computation. The result that there are no new Wieferich primes less than is reported.

     

Bernoulli-Hurwitz numbers, Wieferich primes and Galois representations

Let K be a quadratic imaginary number field with discriminant DK≠−3,−4 and class number one. Fix a prime p?7 which is unramified in K. Given an elliptic curve A/Q with complex multiplication by K, let be the representation which arises from the action of Galois on the Tate module. Herein it is shown that, for all but finitely many inert primes p, the image of a certain deformation of is “as large as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Zp). If p splits in K, then the same result holds as long as a certain Bernoulli-Hurwitz number is a p-adic unit which, in turn, is equivalent to a prime ideal not being a Wieferich place. The proof rests on the theory of elliptic units of Robert and Kubert-Lang, and on the two-variable main conjecture of Iwasawa theory for quadratic imaginary fields.     

Generalizing pairs of complementary sequences and a construction of combinatorial structures

Pairs of complementary sequences such as Golay pairs have zero sum autocorrelation at all non-trivial phases. Several generalizations are known where conditions on either the autocorrelation function, or the entries of the sequences are altered. We aim to unify most of these ideas by introducing autocorrelation functions that apply to any sequences with entries in a set equipped with a ring-like structure which is closed under multiplication and contains multiplicative inverses. Depending on the elements of the chosen set, the resulting complementary pairs may be used to construct a variety of combinatorial structures such as Hadamard matrices, complex generalized weighing matrices, and signed group weighing matrices. We may also construct quasi-cyclic and quasi-constacyclic linear codes which over finite fields of order less than 5 are also Hermitian self-orthogonal. As the literature on binary and ternary Golay sequences is already quite deep, one intention of this paper is to survey and assimilate work on more general pairs of complementary sequences and related constructions of combinatorial objects, and to combine the ideas into a single theoretical framework.     

Infinite co-minimal pairs involving lacunary sequences and generalisations to higher dimensions

In 2011, Nathanson proposed several questions on minimal complements in a group or a semigroup. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the authors. In this article, we study which type of subsets in the integers and free abelian groups of higher rank can be a part of a co-minimal pair. We show that a majority of lacunary sequences have this property. From the conditions established, one can show that any infinite subset of any finitely generated abelian group has uncountably many subsets which is a part of a co-minimal pair. Further, the uncountable collection of sets can be chosen so that they satisfy certain algebraic properties.

     

Enumeration of pairs of sequences by rises,falls and levels

The paper is concerned with the enumeration of pairs of sequences with given specification according to rises, falls and levels. Thus there are nine possibilities RR, ..., LL. Generating functions in the general case are very complicated. However in a number of special cases simple explicit results are obtained.Supported in part by NSF grant GP-37924X.     

On summability of pairs of convolution sequences of probability measures on locally compact semigroups

In this paper we study the problem of convergence in the weak and the vague topology of the sequence

On summability of pairs of convolution sequences of probability measures on locally compact semigroups

In this paper we study the problem of convergence in the weak and the vague topology of the sequence

$$\left( {\mathop \Sigma \limits_{i = 1}^\infty \mathop \Sigma \limits_{j = 1}^\infty a_{ij}^{(n)} \mu ^i *v^j ,n \in \mathbb{N}} \right)$$     

Three‐Phase Barker Arrays

A 3‐phase Barker array is a matrix of third roots of unity for which all out‐of‐phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two‐dimensional 3‐phase Barker arrays have size 2 × 2 or 3 × 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3‐phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a double‐exponentially growing arithmetic function T such that no 3‐phase Barker array of size with exists for all . For example, , , and . When both dimensions are divisible by 3, the existence problem is settled completely: if a 3‐phase Barker array of size exists, then .     

Proof of the Barker array conjecture

Using only elementary methods, we prove Alquaddoomi and Scholtz's conjecture of 1989, that no Barker array having exists except when .

     

Block companion Singer cycles, primitive recursive vector sequences, and coprime polynomial pairs over finite fields

We discuss a conjecture concerning the enumeration of nonsingular matrices over a finite field that are block companion and whose order is the maximum possible in the corresponding general linear group. A special case is proved using some recent results on the probability that a pair of polynomials with coefficients in a finite field is coprime. Connection with an older problem of Niederreiter about the number of splitting subspaces of a given dimension are outlined and an asymptotic version of the conjectural formula is established. Some applications to the enumeration of nonsingular Toeplitz matrices of a given size over a finite field are also discussed.     

Existence of best proximity pairs and equilibrium pairs

In this paper, using the fixed point theorem for Kakutani factorizable multifunctions, we shall prove new existence theorems of best proximity pairs and equilibrium pairs for free abstract economies, which include the previous fixed point theorems and equilibrium existence theorems.