(2,1)-separating systems beyond the probabilistic bound

Building on previous results of Xing, we give new lower bounds on the rate of intersecting codes over large alphabets. The proof is constructive, and uses algebraic geometry, although nothing beyond the basic theory of linear systems on curves. Then, using these new bounds within a concatenation argument, we construct binary (2,1)-separating systems of asymptotic rate exceeding the one given by the probabilistic method, which was the best lower bound available up to now. This answers (negatively) the question of whether this probabilistic bound was exact, which has remained open for more than 30 years. (By the way, we also give a formulation of the separation property in terms of metric convexity, which may be an inspirational source for new research problems.)Comment: Version 7 is a shortened version, so that numbering should match with the journal version (to appear soon). Material on convexity and separation in discrete and continuous spaces has been removed. Readers interested in this material should consult version 6 instea

Some New Bounds For Cover-Free Families Through Biclique Cover

An $(r,w;d)$ cover-free family $(CFF)$ is a family of subsets of a finite set such that the intersection of any $r$ members of the family contains at least $d$ elements that are not in the union of any other $w$ members. The minimum number of elements for which there exists an $(r,w;d)-CFF$ with $t$ blocks is denoted by $N((r,w;d),t)$. In this paper, we show that the value of $N((r,w;d),t)$ is equal to the $d$-biclique covering number of the bipartite graph $I_t(r,w)$ whose vertices are all $w$- and $r$-subsets of a $t$-element set, where a $w$-subset is adjacent to an $r$-subset if their intersection is empty. Next, we introduce some new bounds for $N((r,w;d),t)$. For instance, we show that for $r\geq w$ and $r\geq 2$ $$N((r,w;1),t) \geq c{{r+w\choose w+1}+{r+w-1 \choose w+1}+ 3 {r+w-4 \choose w-2} \over \log r} \log (t-w+1),$$ where $c$ is a constant satisfies the well-known bound $N((r,1;1),t)\geq c\frac{r^2}{\log r}\log t$. Also, we determine the exact value of $N((r,w;d),t)$ for some values of $d$. Finally, we show that $N((1,1;d),4d-1)=4d-1$ whenever there exists a Hadamard matrix of order 4d

Good Random Matrices over Finite Fields

The random matrix uniformly distributed over the set of all m-by-n matrices over a finite field plays an important role in many branches of information theory. In this paper a generalization of this random matrix, called k-good random matrices, is studied. It is shown that a k-good random m-by-n matrix with a distribution of minimum support size is uniformly distributed over a maximum-rank-distance (MRD) code of minimum rank distance min{m,n}-k+1, and vice versa. Further examples of k-good random matrices are derived from homogeneous weights on matrix modules. Several applications of k-good random matrices are given, establishing links with some well-known combinatorial problems. Finally, the related combinatorial concept of a k-dense set of m-by-n matrices is studied, identifying such sets as blocking sets with respect to (m-k)-dimensional flats in a certain m-by-n matrix geometry and determining their minimum size in special cases.Comment: 25 pages, publishe

Identifying codes in vertex-transitive graphs and strongly regular graphs

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs

Asymptotically good binary linear codes with asymptotically good self-intersection spans

If C is a binary linear code, let C^2 be the linear code spanned by intersections of pairs of codewords of C. We construct an asymptotically good family of binary linear codes such that, for C ranging in this family, the C^2 also form an asymptotically good family. For this we use algebraic-geometry codes, concatenation, and a fair amount of bilinear algebra. More precisely, the two main ingredients used in our construction are, first, a description of the symmetric square of an odd degree extension field in terms only of field operations of small degree, and second, a recent result of Garcia-Stichtenoth-Bassa-Beelen on the number of points of curves on such an odd degree extension field.Comment: 18 pages; v2->v3: expanded introduction and bibliography + various minor change

Second-Order Weight Distributions

A fundamental property of codes, the second-order weight distribution, is proposed to solve the problems such as computing second moments of weight distributions of linear code ensembles. A series of results, parallel to those for weight distributions, is established for second-order weight distributions. In particular, an analogue of MacWilliams identities is proved. The second-order weight distributions of regular LDPC code ensembles are then computed. As easy consequences, the second moments of weight distributions of regular LDPC code ensembles are obtained. Furthermore, the application of second-order weight distributions in random coding approach is discussed. The second-order weight distributions of the ensembles generated by a so-called 2-good random generator or parity-check matrix are computed, where a 2-good random matrix is a kind of generalization of the uniformly distributed random matrix over a finite filed and is very useful for solving problems that involve pairwise or triple-wise properties of sequences. It is shown that the 2-good property is reflected in the second-order weight distribution, which thus plays a fundamental role in some well-known problems in coding theory and combinatorics. An example of linear intersecting codes is finally provided to illustrate this fact.Comment: 10 pages, accepted for publication in IEEE Transactions on Information Theory, May 201