Search

Splitter sets are closely related to lattice tilings, and have applications in flash memories and conflict-avoiding codes. The study of k -radius sequences was motivated by some problems occurring in large data transfer. It is observed that the existence of splitter sets yields k -radius sequences of short length. In this paper, we obtain several new results contributing to splitter sets and k -radius sequences. We give some new constructions of perfect splitter sets, as well as some nonexistence results on them. As a byproduct, we obtain some new results on optimal conflict-avoiding codes. Furthermore, we provide several explicit constructions of short k -radius sequences for certain values of n , by establishing the existence of k -additive sequences. In particular, we show that for any fixed k , where fk (n) denotes the shortest length of an n -ary $k$ -radius sequence. This result partially affirms a conjecture posed by Bondy, Lonc, and Rzążewski. [ABSTRACT FROM AUTHOR]

The determination of the cross correlation between an m -sequence and its decimated sequence has been a long-standing research problem. Considering a ternary m -sequence of period 3^3r-1 , we determine the cross correlation distribution for decimations d=3^r+2 and d=3^2r+2 , where \gcd (r,3)=1 . Meanwhile, for a binary m -sequence of period 2^2lm-1 , we make an initial investigation for the decimation d=(2^2lm-1)/(2^m+1)+2^s , where $l\geq 2$ is even and $0\leq s\leq 2m-1$ . It is shown that the cross correlation takes at least four values. Furthermore, we confirm the validity of two famous conjectures due to Sarwate et al. and Helleseth in this case. [ABSTRACT FROM AUTHOR]

Nonuniform group divisible designs (GDDs) have been studied by numerous researchers for the past two decades due to their essential role in the constructions for other types of designs. In this paper, we investigate the existence problem of [ABSTRACT FROM AUTHOR]

A Steiner quadruple system of order v is an ordered pair $${(X, \mathcal{B})}$$, where X is a set of cardinality v, and $${\mathcal{B}}$$ is a set of 4-subsets of X, called blocks, with the property that every 3-subset of X is contained in a unique block. Such designs exist if and only if $${v \equiv 2,4\, (\bmod\, 6)}$$. The first and second proofs of this result were given by Hanani in and in , respectively. All the existing proofs are rather cumbersome, even though simplified proofs have been given by Lenz in and by Hartman in . To study Steiner quadruple systems, Hanani introduced the concept of H-designs in . The purpose of this paper is to provide an alternative existence proof for Steiner quadruple systems via H-designs of type 2. In , Mills showed that for n > 3, n ≠ 5, an H-design of type g exists if and only if ng is even and g( n − 1)( n − 2) is divisible by 3, where the main context is the complicated existence proof for H-designs of type 2. However, Mill's proof was based on the existence result of Steiner quadruple systems. In this paper, by using the theory of candelabra systems and H-frames, we give a new existence proof for H-designs of type 2 independent of the existence result of Steiner quadruple systems. As a consequence, we also provide a new existence proof for Steiner quadruple systems. [ABSTRACT FROM AUTHOR]

Abstract: The whist tournament problem has attracted the interests of many design theorists for over a century. To construct -cyclic whist tournaments, the concept of -cyclic patterned starter whist tournaments was introduced. The research on -cyclic patterned starter whist tournaments dates back to 1954. Although considerable progress has been made in the last decade or so, less has been made about the existence of -cyclic patterned starter whist tournaments, despite the efforts of many authors. In this paper, we will first provide two new nonexistence results for -cyclic patterned starter whist tournaments. Then, we will propose a general recursive construction, i.e., a frame construction, for -cyclic patterned starter whist tournaments. As a consequence, we are able to unify many known constructions for -cyclic patterned starter whist tournaments. The known existence results of such designs are then extended. [Copyright &y& Elsevier]

Motivated by applications in combinatorial group testing for consecutive positives, we investigate a block sequence of a maximum packing MP(t, k, v) which contains the blocks exactly once such that the collection of all blocks together with all unions of two consecutive blocks of this sequence forms an error correcting code with minimum distance d. Such a sequence is usually called a block sequence with consecutive unions having minimum distance d, and denoted by BSCU(t, k, v∣d). In this paper, we show that the necessary conditions for the existence of BSCU(3, 4, v∣4)s of Steiner quadruple systems, namely, v Ξ 2,4 (mod 6) and v ≥ 4, are also sufficient, excepting v = 8, 10. [ABSTRACT FROM AUTHOR]

Recently, Yaakobi et al. introduced codes for b -symbol read channels, where the read operation is performed as a consecutive sequence of b > 2 symbols. In this paper, we establish a Singleton-type bound for b -symbol codes. Codes meeting the Singleton-type bound are called maximum distance separable (MDS) codes, and they are optimal in the sense they attain the maximal minimum b -distance. We introduce a construction method using projective geometry, and then construct several infinite families of linear MDS b -symbol codes over finite fields. The lengths of these codes have a large range. And in some sense, we completely determine the existence of linear MDS b -symbol codes over finite fields for certain parameters. [ABSTRACT FROM AUTHOR]

In this paper, a new class of circulant matrices built from the deterministic filter and the deterministic subsampling is introduced for convolution-based compressed sensing. The pseudo-random sequences are applied in the construction of compressed sensing matrices. By using Katz's and Bombieri's character sum estimation, we are able to design good deterministic compressed sensing matrices for sparse recovery. In the worst case, the sparsity bound in our construction is similar to that of binary compressed sensing matrices constructed by DeVore and partial Fourier matrices constructed by Xu and Xu. Moreover, in the average case, we show that our construction can reconstruct almost all k -sparse vectors with m ≥ O ( k log ⁡ N ) . [ABSTRACT FROM AUTHOR]