Wen-Ching Winnie Li

On the Structure of t-Designs

It is possible to view the combinatorial structures known as (integral) t-designs as

Spectra of regular graphs and hypergraphs and orthogonal polynomials

\mathbb{Z}

Character sums and abelian Ramanujan graphs

-modules in a natural way. In this note we introduce a polynomial associated to each such

A 2-adic approach to the analysis of cyclic codes

\mathbb{Z}

Independence numbers of graphs and generators of ideals

-module. Using this association, we quickly derive explicit bases for the important class of submodules which correspond to the so-called null-designs.

Pseudo-codewords of cycle codes via zeta functions

Abstract In this paper we study the distribution of eigenvalues of regular graphs, regular hypergraphs, and biregular bipartite graphs of given girth by considering the polynomials orthogonal with respect to the measures attached to the spectra of such graphs and to the continuous spectra of their ‘universal covers’. Our estimates are tight for Biggs graphs and generalized polygons. We also give an application to the distribution of eigenvalues of Hecke operators acting on weight 2 cusp forms for certain congruence subgroups.

Hidden Number Problem with the Trace and Bit Security of XTR and LUC

Abstract Let F be a finite field of q elements. In this paper we obtain several estimates on character sums derived from the Riemann hypothesis for curves over F . In particular, we establish an estimate on twisted generalized Kloosterman sums as conjectured by P. Deligne (1977, “Cohomologieetale (SGA 4½),” Lecture Notes in Mathemmatics, Vol. 569, Springer-Verlag, Berlin/Heidelberg/New York) for the case n = 2: |Σ x ∈ N 2 χ( x ) ψ( x )| ≤ 2 q 1/2 for all nontrivial characters (χ, ψ) of N 2 × F 2 . Here F 2 is a quadratic extension of F and N 2 consists of norm 1 (to F ) elements in F 2 . We also present new constructions of Ramanujan graphs based on abelian groups. The character sum estimates are used to prove that these are indeed Ramanujan graphs.

New Optimal Tame Towers of Function Fields over Small Finite Fields

This paper describes how 2-adic numbers can be used to analyze the structure of binary cyclic codes and of cyclic codes defined over Z/sub 2(a)/, a/spl ges/2, the ring of integers modulo 2/sup a/. It provides a 2-adic proof of a theorem of McEliece that characterizes the possible Hamming weights that can appear in a binary cyclic code. A generalization of this theorem is derived that applies to cyclic codes over Z/sub 2(a)/ that are obtained from binary cyclic codes by a sequence of Hensel lifts. This generalization characterizes the number of times a residue modulo 2/sup a/ appears as a component of an arbitrary codeword in the cyclic code. The limit of the sequence of Hensel lifts is a universal code defined over the 2-adic integers. This code was first introduced by Calderbank and Sloane (1995), and is the main subject of this paper. Binary cyclic codes and cyclic codes over Z/sub 2(a)/ are obtained from these universal codes by reduction modulo some power of 2. A special case of particular interest is cyclic codes over Z/sub 4/ that are obtained from binary cyclic codes by means of a single Hensel lift. The binary images of such codes under the Gray isometry include the Kerdock, Preparata, and Delsart-Goethals codes. These are nonlinear binary codes that contain more codewords than any linear code presently known. Fundamental understanding of the composition of codewords in cyclic codes over Z/sub 4/ is central to the search for more families of optimal codes. This paper also constructs even unimodular lattices from the Hensel lift of extended binary cyclic codes that are self-dual with all Hamming weights divisible by 4. The Leech lattice arises in this way as do extremal lattices in dimensions 32 through 48.

Reliabilities of Double-Loop Networks

This article investigates the generators of certain homogeneous ideals which are associated with graphs with bounded independence numbers. These ideals first appeared in the theory oft-designs. The main theorem suggests a new approach to the Clique Problem which isNP-complete. This theorem has a more general form in commutative algebra dealing with ideals associated with unions of linear varieties. This general theorem is stated in the article; a corollary to it generalizes Turán’s theorem on the maximum graphs with a prescribed clique number.

Partition polytopes over 1-dimensional points

Cycle codes are a special case of low-density parity-check (LDPC) codes and as such can be decoded using an iterative message-passing decoding algorithm on the associated Tanner graph. The existence of pseudo-codewords is known to cause the decoding algorithm to fail in certain instances. In this paper, we draw a connection between pseudo-codewords of cycle codes and the so-called edge zeta function of the associated normal graph and show how the Newton polytope of the zeta function equals the fundamental cone of the code, which plays a crucial role in characterizing the performance of iterative decoding algorithms.