Xiande Zhang

Multiply Constant-Weight Codes and the Reliability of Loop Physically Unclonable Functions

We introduce the class of multiply constant-weight codes to improve the reliability of certain physically unclonable function response, and extend classical coding methods to construct multiply constant-weight codes from known \(q\) -ary and constant-weight codes. We derive analogs of Johnson bounds and give constructions showing these bounds to be asymptotically tight up to a constant factor under certain conditions. We also examine the rates of multiply constant-weight codes and demonstrate that these rates are the same as those of constant-weight codes of corresponding parameters.

Universal Cycles for Minimum Coverings of Pairs by Triples, with Application to 2-Radius Sequences

A new ordering, extending the notion of universal cycles of Chung et al. (1992), is proposed for the blocks of k-uniform set systems. Existence of minimum coverings of pairs by triples that possess such an ordering is established for all orders. The application to the construction of short 2-radius sequences is given, along with some new 2-radius sequences found through a computer search.

Existence of resolvable H-designs with group sizes 2, 3, 4 and 6

In 1987, Hartman showed that the necessary condition v ≡ 4 or 8 (mod 12) for the existence of a resolvable SQS(v) is also sufficient for all values of v, with 23 possible exceptions. These last 23 undecided orders were removed by Ji and Zhu in 2005 by introducing the concept of resolvable H-designs. In this paper, we first develop a simple but powerful construction for resolvable H-designs, i.e., a construction of an RH(g2n) from an RH((2g)n), which we call group halving construction. Based on this construction, we provide an alternative existence proof for resolvable SQS(v)s by investigating the existence problem of resolvable H-designs with group size 2. We show that the necessary conditions for the existence of an RH(2n), namely, n ≡ 2 or 4 (mod 6) and n ≥ 4 are also sufficient. Meanwhile, we provide an alternative existence proof for resolvable H-designs with group size 6. These results are obtained by first establishing an existence result for resolvable H-designs with group size 4, that is, the necessary conditions n ≡ 1 or 2 (mod 3) and n ≥ 4 for the existence of an RH(4n) are also sufficient for all values of n except possibly n ∈ {73, 149}. As a consequence, the general existence problem of an RH(gn) is solved leaving mainly the case of g ≡ 0 (mod 12) open. Finally, we show that the necessary conditions for the existence of a resolvable G-design of type gn are also sufficient.

Improved Constructions of Frameproof Codes

Frameproof codes are used to preserve the security in the context of coalition when fingerprinting digital data. Let M_c,l(q) be the largest cardinality of a q-ary c-frameproof code of length l and R_c,l=lim_q→∞ M_c,l(q)/q^{[ l/c]}. It has been determined by Blackburn that R_c,l=1 when l≡1(mod c), R_c,l=2 when c=2 and l is even, and R_3,5=5/3. In this paper, we give a recursive construction for c-frameproof codes of length l with respect to the alphabet size q . As applications of this construction, we establish the existence results for q-ary c-frameproof codes of length c+2 and size c+2/c(q-1)²+1 for all odd q when c=2 and for all q≡4 when c=3 . Furthermore, we show that R_c,c+2=(c+2)/c meeting the upper bound given by Blackburn, for all integers c such that c+1 is a prime power.

Infinite families of optimal splitting authentication codes secure against spoofing attacks of higher order

We consider the problem of constructing optimal authentication codes with splitting. New infinite families of such codes are obtained. In particular, we establish the first known infinite family of optimal authentication codes with splitting that are secure against spoofing attacks of order two.

H-designs with the properties of resolvability or (1, 2)-resolvability

An H-design is said to be (1, α)-resolvable, if its block set can be partitioned into α-parallel classes, each of which contains every point of the design exactly α times. When α = 1, a (1, α)-resolvable H-design of type gn is simply called a resolvable H-design and denoted by RH(gn), for which the general existence problem has been determined leaving mainly the case of g ≡ 0 (mod 12) open. When α = 2, a (1, 2)-RH(1n) is usually called a (1, 2)-resolvable Steiner quadruple system of order n, for which the existence problem is far from complete. In this paper, we consider these two outstanding problems. First, we prove that an RH(12n) exists for all n ≥ 4 with a small number of possible exceptions. Next, we give a near complete solution to the existence problem of (1, 2)-resolvable H-designs with group size 2. As a consequence, we obtain a near complete solution to the above two open problems.

Constructions of Optimal and Near-Optimal Multiply Constant-Weight Codes

Multiply constant-weight codes (MCWCs) have been recently studied to improve the reliability of certain physically unclonable function response. In this paper, we give combinatorial constructions for the MCWCs, which yield several new infinite families of optimal MCWCs. Furthermore, we demonstrate that the Johnson-type upper bounds of the MCWCs are asymptotically tight for fixed Hamming weights and distances. Finally, we provide bounds and constructions of the 2-D MCWCs.

Optimal Ternary Constant-Weight Codes With Weight 4 and Distance 5

Constant-weight codes (CWCs) play an important role in coding theory. The problem of determining the sizes for optimal ternary CWCs with length <i>n</i>, weight 4, and minimum Hamming distance 5 ((<i>n</i>,5,4)₃ code) has been settled for all positive integers <i>n</i> ≤ 10 or <i>n</i> >; 10 and <i>n</i> ≡ 1 mod 3 with <i>n</i> ∈ {13,52,58} undetermined. In this paper, we investigate the problem of constructing optimal (<i>n</i>,5,4)₃ codes for all lengths <i>n</i> with the tool of group divisible codes. We determine the size of an optimal (<i>n</i>,5,4)₃ code for each integer <i>n</i> ≥ 4 leaving the lengths <i>n</i> ∈ {12,13,21,27,33,39,45,52} unsolved.

Decompositions of edge-colored digraphs: A new technique in the construction of constant-weight codes and related families

We demonstrate that certain Johnson-type bounds are asymptotically exact for a variety of classes of codes, namely, constant-composition codes, nonbinary constant-weight codes and multiply constant-weight codes. This was achieved via an interesting application of the theory of decomposition of edge-colored digraphs.

The α-Arboricity of Complete Uniform Hypergraphs

α-acyclicity is an important notion in database theory. The α-arboricity of a hypergraph ℋ is the minimum number of α-acyclic hypergraphs that partition the edge set of ℋ. The α-arboricity of the complete 3-uniform hypergraph is determined completely.