Maximal mm-distance sets containing the representation of the Hamming graph H(n,m)H(n,m)

A set $X$ in the Euclidean space $\mathbb{R}^d$ is called an $m$-distance set if the set of Euclidean distances between two distinct points in $X$ has size $m$. An $m$-distance set $X$ in $\mathbb{R}^d$ is said to be maximal if there does not exist a vector $x$ in $\mathbb{R}^d$ such that the union of $X$ and $\{x\}$ still has only $m$ distances. Bannai--Sato--Shigezumi (2012) investigated the maximal $m$-distance sets which contain the Euclidean representation of the Johnson graph $J(n,m)$. In this paper, we consider the same problem for the Hamming graph $H(n,m)$. The Euclidean representation of $H(n,m)$ is an $m$-distance set in $\mathbb{R}^{m(n-1)}$. We prove that the maximum $n$ is $m^2 + m - 1$ such that the representation of $H(n,m)$ is not maximal as an $m$-distance set. Moreover we classify the largest $m$-distance sets which contain the representation of $H(n,m)$ for $m\leq 4$ and any $n$. We also classify the maximal $2$-distance sets in $\mathbb{R}^{2n-1}$ which contain the representation of $H(n,2)$ for any $n$.Comment: 19 pages, no figur

Combinatorial Toolbox for Protein Sequence Design and Landscape Analysis in the Grand Canonical Model

In modern biology, one of the most important research problems is to understand how protein sequences fold into their native 3D structures. To investigate this problem at a high level, one wishes to analyze the protein landscapes, i.e., the structures of the space of all protein sequences and their native 3D structures. Perhaps the most basic computational problem at this level is to take a target 3D structure as input and design a fittest protein sequence with respect to one or more fitness functions of the target 3D structure. We develop a toolbox of combinatorial techniques for protein landscape analysis in the Grand Canonical model of Sun, Brem, Chan, and Dill. The toolbox is based on linear programming, network flow, and a linear-size representation of all minimum cuts of a network. It not only substantially expands the network flow technique for protein sequence design in Kleinberg's seminal work but also is applicable to a considerably broader collection of computational problems than those considered by Kleinberg. We have used this toolbox to obtain a number of efficient algorithms and hardness results. We have further used the algorithms to analyze 3D structures drawn from the Protein Data Bank and have discovered some novel relationships between such native 3D structures and the Grand Canonical model

Pseudocodewords of Tanner graphs

This papers presents a detailed analysis of pseudocodewords of Tanner graphs. Pseudocodewords arising on the iterative decoder's computation tree are distinguished from pseudocodewords arising on finite degree lifts. Lower bounds on the minimum pseudocodeword weight are presented for the BEC, BSC, and AWGN channel. Some structural properties of pseudocodewords are examined, and pseudocodewords and graph properties that are potentially problematic with min-sum iterative decoding are identified. An upper bound on the minimum degree lift needed to realize a particular irreducible lift-realizable pseudocodeword is given in terms of its maximal component, and it is shown that all irreducible lift-realizable pseudocodewords have components upper bounded by a finite value $t$ that is dependent on the graph structure. Examples and different Tanner graph representations of individual codes are examined and the resulting pseudocodeword distributions and iterative decoding performances are analyzed. The results obtained provide some insights in relating the structure of the Tanner graph to the pseudocodeword distribution and suggest ways of designing Tanner graphs with good minimum pseudocodeword weight.Comment: To appear in Nov. 2007 issue of IEEE Transactions on Information Theor

Distance-2 MDS codes and latin colorings in the Doob graphs

The maximum independent sets in the Doob graphs D(m,n) are analogs of the distance-2 MDS codes in Hamming graphs and of the latin hypercubes. We prove the characterization of these sets stating that every such set is semilinear or reducible. As related objects, we study vertex sets with maximum cut (edge boundary) in D(m,n) and prove some facts on their structure. We show that the considered two classes (the maximum independent sets and the maximum-cut sets) can be defined as classes of completely regular sets with specified 2-by-2 quotient matrices. It is notable that for a set from the considered classes, the eigenvalues of the quotient matrix are the maximum and the minimum eigenvalues of the graph. For D(m,0), we show the existence of a third, intermediate, class of completely regular sets with the same property.Comment: v2: revised; accepted versio

Eigenvalues of subgraphs of the cube

We consider the problem of maximising the largest eigenvalue of subgraphs of the hypercube $Q_d$ of a given order. We believe that in most cases, Hamming balls are maximisers, and our results support this belief. We show that the Hamming balls of radius $o(d)$ have largest eigenvalue that is within $1 + o(1)$ of the maximum value. We also prove that Hamming balls with fixed radius maximise the largest eigenvalue exactly, rather than asymptotically, when $d$ is sufficiently large. Our proofs rely on the method of compressions.Comment: 27 page

Maximal 22-distance sets containing the regular simplex

A finite subset $X$ of the Euclidean space is called an $m$-distance set if the number of distances between two distinct points in $X$ is equal to $m$. An $m$-distance set $X$ is said to be maximal if any vector cannot be added to $X$ while maintaining the $m$-distance condition. We investigate a necessary and sufficient condition for vectors to be added to a regular simplex such that the set has only $2$ distances. We construct several $d$-dimensional maximal $2$-distance sets that contain a $d$-dimensional regular simplex. In particular, there exist infinitely many maximal non-spherical $2$-distance sets that contain both the regular simplex and the representation of a strongly resolvable design. The maximal $2$-distance set has size $2s^2(s+1)$, and the dimension is $d=(s-1)(s+1)^2-1$, where $s$ is a prime power.Comment: 15 pages, no figur

Edge Coloring and Stopping Sets Analysis in Product Codes with MDS components

We consider non-binary product codes with MDS components and their iterative row-column algebraic decoding on the erasure channel. Both independent and block erasures are considered in this paper. A compact graph representation is introduced on which we define double-diversity edge colorings via the rootcheck concept. An upper bound of the number of decoding iterations is given as a function of the graph size and the color palette size $M$. Stopping sets are defined in the context of MDS components and a relationship is established with the graph representation. A full characterization of these stopping sets is given up to a size $(d_1+1)(d_2+1)$, where $d_1$ and $d_2$ are the minimum Hamming distances of the column and row MDS components respectively. Then, we propose a differential evolution edge coloring algorithm that produces colorings with a large population of minimal rootcheck order symbols. The complexity of this algorithm per iteration is $o(M^{\aleph})$, for a given differential evolution parameter $\aleph$, where $M^{\aleph}$ itself is small with respect to the huge cardinality of the coloring ensemble. The performance of MDS-based product codes with and without double-diversity coloring is analyzed in presence of both block and independent erasures. In the latter case, ML and iterative decoding are proven to coincide at small channel erasure probability. Furthermore, numerical results show excellent performance in presence of unequal erasure probability due to double-diversity colorings.Comment: 82 pages, 14 figures, and 4 tables, Submitted to the IEEE Transactions on Information Theory, Dec. 2015, paper IT-15-110

Submodular meets Structured: Finding Diverse Subsets in Exponentially-Large Structured Item Sets

To cope with the high level of ambiguity faced in domains such as Computer Vision or Natural Language processing, robust prediction methods often search for a diverse set of high-quality candidate solutions or proposals. In structured prediction problems, this becomes a daunting task, as the solution space (image labelings, sentence parses, etc.) is exponentially large. We study greedy algorithms for finding a diverse subset of solutions in structured-output spaces by drawing new connections between submodular functions over combinatorial item sets and High-Order Potentials (HOPs) studied for graphical models. Specifically, we show via examples that when marginal gains of submodular diversity functions allow structured representations, this enables efficient (sub-linear time) approximate maximization by reducing the greedy augmentation step to inference in a factor graph with appropriately constructed HOPs. We discuss benefits, tradeoffs, and show that our constructions lead to significantly better proposals

Pattern Recognition on Oriented Matroids: Symmetric Cycles in the Hypercube Graphs

If V is the vertex sequence of a symmetric 2t-cycle in the hypercube graph with the vertices {1,-1}^t, then for any vertex T of the graph there exists a unique inclusion-minimal subset of V such that T is the sum of its elements. We present a simple combinatorial statistic on decompositions of vertices of the hypercube graphs with respect to symmetric cycles and describe their basic metric properties.Comment: 10 pages; v.2,3 - minor improvements; v.4 - appendix and references adde

Graph powers, Delsarte, Hoffman, Ramsey and Shannon

The $k$-th $p$-power of a graph $G$ is the graph on the vertex set $V(G)^k$, where two $k$-tuples are adjacent iff the number of their coordinates which are adjacent in $G$ is not congruent to 0 modulo $p$. The clique number of powers of $G$ is poly-logarithmic in the number of vertices, thus graphs with small independence numbers in their $p$-powers do not contain large homogenous subsets. We provide algebraic upper bounds for the asymptotic behavior of independence numbers of such powers, settling a conjecture of Alon and Lubetzky up to a factor of 2. For precise bounds on some graphs, we apply Delsarte's linear programming bound and Hoffman's eigenvalue bound. Finally, we show that for any nontrivial graph $G$, one can point out specific induced subgraphs of large $p$-powers of $G$ with neither a large clique nor a large independent set. We prove that the larger the Shannon capacity of $\bar{G}$ is, the larger these subgraphs are, and if $G$ is the complete graph, then some $p$-power of $G$ matches the bounds of the Frankl-Wilson Ramsey construction, and is in fact a subgraph of a variant of that construction