Chaoping Xing

Linear authentication codes: bounds and constructions

In this paper, we consider a new class of unconditionally secure authentication codes, called linear authentication codes (or linear A-codes). We show that a linear A-code can be characterized by a family of subspaces of a vector space over a finite field. We then derive an upper bound on the size of the source space when other parameters of the system, that is, the sizes of the key space and the authenticator space, and the deception probability, are fixed. We give constructions that are asymptotically close to the bound and show applications of these codes in constructing distributed authentication systems.

On Self-Dual Cyclic Codes Over Finite Fields

In coding theory, self-dual codes and cyclic codes are important classes of codes which have been extensively studied. The main objects of study in this paper are self-dual cyclic codes over finite fields, i.e., the intersection of these two classes. We show that self-dual cyclic codes of length <i>n</i> over \BBF<i>q</i> exist if and only if <i>n</i> is even and <i>q</i> = 2<i>m</i> with <i>m</i> a positive integer. The enumeration of such codes is also investigated. When <i>n</i> and <i>q</i> are even, there is always a trivial self-dual cyclic code with generator polynomial <i>x</i>ⁿ/₂+1. We, therefore, classify the existence of self-dual cyclic codes, for given <i>n</i> and <i>q</i> , into two cases: when only the trivial one exists and when two or more such codes exist. Given <i>n</i> and <i>m</i> , an easy criterion to determine which of these two cases occurs is given in terms of the prime factors of <i>n</i>, for most <i>n</i> . We also show that, over a fixed field, the latter case occurs more frequently as the length grows.

Asymptotic bounds on quantum codes from algebraic geometry codes

We generalize a characterization of p-ary (p is a prime) quantum codes given by Feng and Xing to q-ary (q is a prime power) quantum codes. This characterization makes it possible to convert an asymptotic bound of Stichtenoth and Xing for nonlinear algebraic geometry codes to a quantum asymptotic bound. Besides, we also investigate the asymptotic behavior of quantum codes

Nets, ( t, s )-Sequences, and Algebraic Geometry

The star discrepancy is a classical measure for the irregularity of distribution of finite sets and infinite sequences of points in the s-dimensional unit cube P = [0, 1]8. Point sets and sequences with small star discrepancy in I8 are informally called low-discrepancy point sets, respectively low-discrepancy sequences, in I. It is also customary to speak of sets, respectively sequences, of quasirandom points in 78. Such point sets and sequences play a crucial role in applications of numerical quasi-Monte Carlo methods. In fact, the efficiency of a quasi-Monte Carlo method depends to a significant extent on the quality of the quasirandom points that are employed, i.e., on how small their star discrepancy is. Therefore, it is a matter of considerable interest to devise techniques for the construction of point sets and sequences with as small a star discrepancy as possible. The reader who desires more background on discrepancy theory and quasi-Monte Carlo methods is referred to the books of Hua and Wang [9], Kuipers and Niederreiter [10], and Niederreiter[21], the survey article of Niederreiter [16], and the recent monograph of Drmota and Tichy [4].

List decoding reed-solomon, algebraic-geometric, and gabidulin subcodes up to the singleton bound

We consider Reed-Solomon (RS) codes whose evaluation points belong to a subfield, and give a linear-algebraic list decoding algorithm that can correct a fraction of errors approaching the code distance, while pinning down the candidate messages to a well-structured affine space of dimension a constant factor smaller than the code dimension. By pre-coding the message polynomials into a subspace-evasive set, we get a Monte Carlo construction of a subcode of Reed-Solomon codes that can be list decoded from a fraction (1-R-ε) of errors in polynomial time (for any fixed ε > 0) with a list size of O(1/ε). Our methods extend to algebraic-geometric (AG) codes, leading to a similar claim over constant-sized alphabets. This matches parameters of recent results based on folded variants of RS and AG codes. but our construction here gives subcodes of Reed-Solomon and AG codes themselves (albeit with restrictions on the evaluation points). Further, the underlying algebraic idea also extends nicely to Gabidulins construction of rank-metric codes based on linearized polynomials. This gives the first construction of positive rate rank-metric codes list decodable beyond half the distance, and in fact gives codes of rate R list decodable up to the optimal (1-R-ε) fraction of rank errors. A similar claim holds for the closely related subspace codes studied by Koetter and Kschischang. We introduce a new notion called subspace designs as another way to pre-code messages and prune the subspace of candidate solutions. Using these, we also get a deterministic construction of a polynomial time list decodable subcode of RS codes. By using a cascade of several subspace designs, we extend our approach to AG codes, which gives the first deterministic construction of an algebraic code family of rate R with efficient list decoding from 1-R-ε fraction of errors over an alphabet of constant size (that depends only on ε). The list size bound is almost a constant (governed by log* (block length)), and the code can be constructed in quasi-polynomial time.

Asymmetric Quantum Codes: Characterization and Constructions

The stabilizer method for constructing a class of asymmetric quantum codes (AQC), called additive AQC, has been established by Aly et.al. In this paper, we present a new characterization of AQC, which generalizes a result of the symmetric case known previously. As an application of the characterization, we establish a relationship of AQC with classical error-correcting codes and show a few examples of good AQC with specific parameters. By using this relationship, we obtain an asymptotic bound on AQCs from algebraic geometry codes.

Application of Classical Hermitian Self-Orthogonal MDS Codes to Quantum MDS Codes

In this paper, we first construct several classes of classical Hermitian self-orthogonal maximum distance separable (MDS) codes. Through these classical codes, we are able to obtain various quantum MDS codes. It turns out that many of our quantum codes are new in the sense that the parameters of our quantum codes cannot be obtained from all previous constructions.

Coding and Cryptology

An Infinite Class of Balanced Vectorial Boolean Functions with Optimum Algebraic Immunity and Good Nonlinearity.- Separation and Witnesses.- Binary Covering Arrays and Existentially Closed Graphs.- A Class of Three-Weight and Four-Weight Codes.- Equal-Weight Fingerprinting Codes.- Problems on Two-Dimensional Synchronization Patterns.- A New Client-to-Client Password-Authenticated Key Agreement Protocol.- Elliptic Twin Prime Conjecture.- Hunting for Curves with Many Points.- List Decoding of Binary Codes-A Brief Survey of Some Recent Results.- Recent Developments in Low-Density Parity-Check Codes.- On the Applicability of Combinatorial Designs to Key Predistribution for Wireless Sensor Networks.- On Weierstrass Semigroups of Some Triples on Norm-Trace Curves.- ERINDALE: A Polynomial Based Hashing Algorithm.- A Survey of Algebraic Unitary Codes.- New Family of Non-Cartesian Perfect Authentication Codes.- On the Impossibility of Strong Encryption Over .- Minimum Distance between Bent and Resilient Boolean Functions.- Unconditionally Secure Approximate Message Authentication.- Multiplexing Realizations of the Decimation-Hadamard Transform of Two-Level Autocorrelation Sequences.- On Cayley Graphs, Surface Codes, and the Limits of Homological Coding for Quantum Error Correction.

Quantum codes from concatenated algebraic-geometric codes

We apply Steanes enlargement of the Calderbank-Shor-Steane (CSS) codes and additive codes over F/sub 4/ to concatenated algebraic-geometric codes to construct many good quantum codes with fewer restrictions on the parameters compared to some known quantum codes. Some of the quantum codes we have constructed are either optimal or have parameters as good as the best known codes, while some have parameters better than those obtained from other known constructions.

Nonlinear codes from algebraic curves improving the Tsfasman-Vladut-Zink bound

In the present paper, we construct a class of nonlinear codes by making use of higher order derivatives of certain functions of algebraic curves. It turns out that the asymptotic bound derived from the Goppa geometry codes can be improved for the entire interval (0,1). In particular, the Tsfasman-Vladut-Zink (TVZ) bound is ameliorated for the entire interval (0,1).