Ka Hin Leung

Cyclic codes over GR(p2,m) of length pk

We study cyclic codes of length p^k over GR(p^2,m), or equivalently, ideals of the ring GR(p^2,m)/. We derive a method of representing the ideals, and classify all ideals in the ring GR(p^2,m)/. We also analyse the duals, and identify the self-dual ideals.

The Field Descent Method

We obtain a broadly applicable decomposition of group ring elements into a “subfield part” and a “kernel part”. Applications include the verification of Lander’s conjecture for all difference sets whose order is a power of a prime >3 and for all McFarland, Spence and Chen/Davis/Jedwab difference sets. We obtain a new general exponent bound for difference sets. We show that there is no circulant Hadamard matrix of order v with 4<v<548, 964, 900 and no Barker sequence of length l with 13 < l ≤ 1022.

New restrictions on possible orders of circulant Hadamard matrices

We obtain several new number theoretic results which improve the field descent method. We use these results to rule out many of the known open cases of the circulant Hadamard matrix conjecture. In particular, the only known open case of the Barker sequence conjecture is settled.

Partial Difference Triples

AbstractIt is known that a strongly regular semi-Cayley graph (with respect to a group G) corresponds to a triple of subsets (C, D, D′) of G. Such a triple (C, D, D′) is called a partial difference triple. First, we study the case when D ∪ D′ is contained in a proper normal subgroup of G. We basically determine all possible partial difference triples in this case. In fact, when

Binary cyclic codes with two primitive nonzeros

|G| \ne 8

Nonexistence of abelian difference sets: Lander’s conjecture for prime power orders

nor 25, all partial difference triples come from a certain family of partial difference triples. Second, we investigate partial difference triples over cyclic group. We find a few nontrivial examples of strongly regular semi-Cayley graphs when |G| is even. This gives a negative answer to a problem raised by de Resmini and Jungnickel. Furthermore, we determine all possible parameters when G is cyclic. Last, as an application of the theory of partial difference triples, we prove some results concerned with strongly regular Cayley graphs.

Constructions of Relative Difference Sets with Classical Parameters and Circulant Weighing Matrices

We show that a circulant weighing matrix of order n and weight 16 exists if and only if n>=21 and n is a multiple of 14,21 or 31.

New Hadamard matrices of order 4 p 2 obtained from Jacobi sums of order 16

In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of Tr(xd) over