Asymmetric binary covering codes

An asymmetric binary covering code of length n and radius R is a subset C of the n-cube Q_n such that every vector x in Q_n can be obtained from some vector c in C by changing at most R 1's of c to 0's, where R is as small as possible. K^+(n,R) is defined as the smallest size of such a code. We show K^+(n,R) is of order 2^n/n^R for constant R, using an asymmetric sphere-covering bound and probabilistic methods. We show K^+(n,n-R')=R'+1 for constant coradius R' iff n>=R'(R'+1)/2. These two results are extended to near-constant R and R', respectively. Various bounds on K^+ are given in terms of the total number of 0's or 1's in a minimal code. The dimension of a minimal asymmetric linear binary code ([n,R]^+ code) is determined to be min(0,n-R). We conclude by discussing open problems and techniques to compute explicit values for K^+, giving a table of best known bounds.Comment: 16 page

On the Peak-to-Mean Envelope Power Ratio of Phase-Shifted Binary Codes

The peak-to-mean envelope power ratio (PMEPR) of a code employed in orthogonal frequency-division multiplexing (OFDM) systems can be reduced by permuting its coordinates and by rotating each coordinate by a fixed phase shift. Motivated by some previous designs of phase shifts using suboptimal methods, the following question is considered in this paper. For a given binary code, how much PMEPR reduction can be achieved when the phase shifts are taken from a 2^h-ary phase-shift keying (2^h-PSK) constellation? A lower bound on the achievable PMEPR is established, which is related to the covering radius of the binary code. Generally speaking, the achievable region of the PMEPR shrinks as the covering radius of the binary code decreases. The bound is then applied to some well understood codes, including nonredundant BPSK signaling, BCH codes and their duals, Reed-Muller codes, and convolutional codes. It is demonstrated that most (presumably not optimal) phase-shift designs from the literature attain or approach our bound.Comment: minor revisions, accepted for IEEE Trans. Commun

Partial-sum queries in OLAP data cubes using covering codes

A partial-sum query obtains the summation over a set of specified cells of a data cube. We establish a connection between the covering problem in the theory of error-correcting codes and the partial-sum problem and use this connection to devise algorithms for the partial-sum problem with efficient space-time trade-offs. For example, using our algorithms, with 44 percent additional storage, the query response time can be improved by about 12 percent; by roughly doubling the storage requirement, the query response time can be improved by about 34 percent

On kissing numbers and spherical codes in high dimensions

We prove a lower bound of $\Omega (d^{3/2} \cdot (2/\sqrt{3})^d)$ on the kissing number in dimension $d$. This improves the classical lower bound of Chabauty, Shannon, and Wyner by a linear factor in the dimension. We obtain a similar linear factor improvement to the best known lower bound on the maximal size of a spherical code of acute angle $\theta$ in high dimensions

On upper bounds on the smallest size of a saturating set in a projective plane

In a projective plane $\Pi _{q}$ (not necessarily Desarguesian) of order $q,$ a point subset $S$ is saturating (or dense) if any point of $\Pi _{q}\setminus S$ is collinear with two points in$~S$. Using probabilistic methods, the following upper bound on the smallest size $s(2,q)$ of a saturating set in $\Pi _{q}$ is proved: \begin{equation*} s(2,q)\leq 2\sqrt{(q+1)\ln (q+1)}+2\thicksim 2\sqrt{q\ln q}. \end{equation*} We also show that for any constant $c\ge 1$ a random point set of size $k$ in $\Pi _{q}$ with $2c\sqrt{(q+1)\ln(q+1)}+2\le k<\frac{q^{2}-1}{q+2}\thicksim q$ is a saturating set with probability greater than $1-1/(q+1)^{2c^{2}-2}.$ Our probabilistic approach is also applied to multiple saturating sets. A point set $S\subset \Pi_{q}$ is $(1,\mu)$-saturating if for every point $Q$ of $\Pi _{q}\setminus S$ the number of secants of $S$ through $Q$ is at least $\mu$, counted with multiplicity. The multiplicity of a secant $\ell$ is computed as ${\binom{{\#(\ell \,\cap S)}}{{2}}}.$ The following upper bound on the smallest size $s_{\mu }(2,q)$ of a $(1,\mu)$-saturating set in $\Pi_{q}$ is proved: \begin{equation*} s_{\mu }(2,q)\leq 2(\mu +1)\sqrt{(q+1)\ln (q+1)}+2\thicksim 2(\mu +1)\sqrt{ q\ln q}\,\text{ for }\,2\leq \mu \leq \sqrt{q}. \end{equation*} By using inductive constructions, upper bounds on the smallest size of a saturating set (as well as on a $(1,\mu)$-saturating set) in the projective space $PG(N,q)$ are obtained. All the results are also stated in terms of linear covering codes.Comment: 15 pages, 24 references, misprints are corrected, Sections 3-5 and some references are adde