An update on primitive ternary complementary pairs

In two recent papers we overhauled the theory of ternary complementary pairs, focusing on questions relating to the possible weights of pairs, and special pairs from which all others can be derived, which we call “primitive.”Of particular interest at this time is a new refinement of the concept of primitivity, which necessitates some revisions to our tables. In this article we report on the state of the art with respect to primitive pairs and elaborate on some conjectures in light of new data.30 new primitive pairs are given; the status of 12 previously “primitive” pairs is changed to “imprimitive.”     

A complete description of Golay pairs for lengths up to 100

In his 1961 paper, Marcel Golay showed how the search for pairs of binary sequences of length with complementary autocorrelation is at worst a problem. Andres, in his 1977 master's thesis, developed an algorithm which reduced this to a search and investigated lengths up to 58 for existence of pairs. In this paper, we describe refinements to this algorithm, enabling a search at length 82. We find no new pairs at the outstanding lengths 74 and 82. In extending the theory of composition, we are able to obtain a closed formula for the number of pairs of length generated by a primitive pair of length . Combining this with the results of searches at all allowable lengths up to 100, we identify five primitive pairs. All others pairs of lengths less than 100 may be derived using the methods outlined.

     

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.     

On overpartition pairs into odd parts modulo powers of 2

In 2012, Lin (Electron. J. Combin. 19(2) (2012) #P17) investigated the 2 and 3-divisibility properties for ${\overline{pp}}_o\left(n\right)$, the number of overpartition pairs into odd parts. Using modular forms, he proved that for a fixed positive integer $k$, ${\overline{pp}}_o\left(n\right)$ is almost always divisible by $2^k$. In this paper, we prove several congruences for ${\overline{pp}}_o\left(n\right)$ modulo higher powers of 2 in an elementary way.     

A trace representation of binary Jacobi sequences

We determine the trace function representation, or equivalently, the Fourier spectral sequences of binary Jacobi sequences of period pq, where p and q are two distinct odd primes. This includes the twin-prime sequences of period p(p+2) whenever both p and p+2 are primes, corresponding to cyclic Hadamard difference sets.