Column Permutations Research Articles

Enhancing Iterative Decoding of Cyclic LDPC Codes Using Their Automorphism Groups

For cyclic LDPC codes, we propose to use their automorphism groups to improve the iterative decoding performance. The basic idea is to construct nonequivalent parity-check matrices via column permutations. Three types of iterative decoders are devised to take advantage of the code's automorphism group. In this paper we focus on cyclic LDPC codes defined by a circulant parity-check matrix and consider two known subgroups of the automorphism group of a cyclic code. For the large class of idempotent-based cyclic LDPC codes in the literature, we show that the two subgroups only provide equivalent parity-check matrices and thus cannot be harnessed for iterative decoding. Towards exploiting the automorphism group of a code, we propose a new class of cyclic LDPC codes based on pseudo-cyclic MDS codes with two information symbols, for which nonequivalent parity-check matrices are obtained. Simulation results show that for our constructed codes of short lengths, the automorphism group can significantly enhance the iterative decoding performance.

A Survey of Forbidden Configuration Results

Let $F$ be a $k\times \ell$ (0,1)-matrix. We say a (0,1)-matrix $A$ has $F$ as a configuration if there is a submatrix of $A$ which is a row and column permutation of $F$. In the language of sets, a configuration is a trace and in the language of hypergraphs a configuration is a subhypergraph.Let $F$ be a given $k\times \ell$ (0,1)-matrix. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. The matrix $F$ need not be simple. We define $\hbox{forb}(m,F)$ as the maximum number of columns of any simple $m$-rowed matrix $A$ which do not contain $F$ as a configuration. Thus if $A$ is an $m\times n$ simple matrix which has no submatrix which is a row and column permutation of $F$ then $n\le\hbox{forb}(m,F)$. Or alternatively if $A$ is an $m\times (\hbox{forb}(m,F)+1)$ simple matrix then $A$ has a submatrix which is a row and column permutation of $F$. We call $F$ a forbidden configuration. The fundamental result is due to Sauer, Perles and Shelah, Vapnik and Chervonenkis. For $K_k$ denoting the $k\times 2^k$ submatrix of all (0,1)-columns on $k$ rows, then $\hbox{forb}(m,K_k)=\binom{m}{k-1}+\binom{m}{k-2}+\cdots \binom{m}{0}$. We seek asymptotic results for $\hbox{forb}(m,F)$ for a fixed $F$ and as $m$ tends to infinity . A conjecture of Anstee and Sali predicts the asymptotically best constructions from which to derive the asymptotics of $\hbox{forb}(m,F)$. The conjecture has helped guide the research and has been verified for $k\times \ell$ $F$ with $k=1,2,3$ and for simple $F$ with $k=4$ as well as other cases including $\ell=1,2$. We also seek exact values for $\hbox{forb}(m,F)$.

Using sampled information: is it enough for the sparse matrix–vector product locality optimization?

SUMMARYOne of the main factors that affect the performance of the sparse matrix–vector product (SpMV) is the low data reuse caused by the irregular and indirect memory access patterns. Different strategies to deal with this problem such as data reordering techniques have been proposed. The computational cost of these techniques is typically high because they consider all the nonzeros of the sparse matrix in order to find an appropriate permutation of rows and columns that improves the SpMV performance. In this paper, we analyze the possibility of increasing the locality of the SpMV using incomplete information in the reordering process. This partial information comes as a consequence of considering only a subset of the nonzero elements of the matrix. These nonzeros are obtained from the original matrix through a sampling process. In particular, two different sampling methods have been considered: a random sampling and an event‐based sampling using hardware counters. We have detected that a small number of samples is enough to obtain quality reorderings. As a consequence, using sampling‐based reorderings leads to noticeable performance improvements with respect to the non‐reordered matrices, reaching speedup values up to 2.1 × . In addition, an important reduction in the computational time required by the reordering technique has been observed. Copyright © 2012 John Wiley & Sons, Ltd.

Hadamard Matrices of Order 32

AbstractTwo Hadamard matrices are considered equivalent if one is obtained from the other by a sequence of operations involving row or column permutations or negations. We complete the classification of Hadamard matrices of order 32. It turns out that there are exactly 13,710,027 such matrices up to equivalence.

Nestedness and τ-temperature in ecological networks

The τ-temperature is a measure of disorder of bipartite networks that is based on the total Manhattan distance of the adjacency matrix. Two properties of this measure are that it does not depend on permutations of lines or columns that have the same connectivity and it is completely determined by connectivities of lines and columns. The normalisation of τ is done by an uniform random matrix whose elements were previously sorted. τ shows no bias against uniform random matrices of several occupations, ρ, sizes, L, and shapes. The scaling of the total Manhattan distance of a random matrix is Drand∝L3ρ while the same scaling for a full nested matrix is Dnest∝L3ρ3/2. We test τ for a large set of empirical matrices to verify these scalings. The index τ correlates better with the temperature of Atmar than with the NODF index of nestedness. We conclude this work by discussing differences between nestedness indices and order/disorder indices.

Forbidden configurations and Steiner designs

Let $${\mathcal{F}}$$ be a (0, 1) matrix. A (0, 1) matrix $${\mathcal{M}}$$ is said to have $${\mathcal{F}}$$ as a configuration if there is a submatrix of $${\mathcal{M}}$$ which is a row and column permutation of $${\mathcal{F}}$$ . We say that a matrix $${\mathcal{M}}$$ is simple if it has no repeated columns. For a given $${v \in \mathbb{N}}$$ , we shall denote by forb $${(v, \mathcal{F})}$$ the maximum number of columns in a simple (0, 1) matrix with v rows for which $${\mathcal{F}}$$ does not occur as a configuration. We say that a matrix $${\mathcal{M}}$$ is maximal for $${\mathcal{F}}$$ if $${\mathcal{M}}$$ has forb $${(v, \mathcal{F})}$$ columns. In this paper we show that for certain natural choices of $${\mathcal{F}}$$ , forb $${(v, \mathcal{F})\leq\frac{\binom{v}{t}}{t+1}}$$ . In particular this gives an extremal characterization for Steiner t-designs as maximal (0, 1) matrices in terms of certain forbidden configurations.

On Minimum Saturated Matrices

Motivated both by the work of Anstee, Griggs, and Sali on forbidden submatrices and also by the extremal sat-function for graphs, we introduce sat-type problems for matrices. Let $${\mathcal{F}}$$ be a family of k-row matrices. A matrix M is called $${\mathcal{F}}$$ -admissible if M contains no submatrix $${F \in \mathcal{F}}$$ (as a row and column permutation of F). A matrix M without repeated columns is $${\mathcal{F}}$$ -saturated if M is $${\mathcal{F}}$$ -admissible but the addition of any column not present in M violates this property. In this paper we consider the function sat( $${n, \mathcal{F}}$$ ) which is the minimal number of columns of an $${\mathcal{F}}$$ -saturated matrix with n rows. We establish the estimate sat $${(n, \mathcal{F})=O(n^{k-1})}$$ for any family $${\mathcal{F}}$$ of k-row matrices and also compute the sat-function for a few small forbidden matrices.

A matrix-theoretic approach for analyzing quasi-cyclic low-density parity-check codes

A matrix-theoretic approach for studying quasi-cyclic codes based on matrix transformations via Fourier transforms and row and column permutations is developed. These transformations put a parity-check matrix in the form of an array of circulant matrices into a diagonal array of matrices of the same size over an extension field. The approach is amicable to the analysis and construction of quasi-cyclic low-density parity-check codes since it takes into account the specific parity-check matrix used for decoding with iterative message-passing algorithms. Based on this approach, the dimension of the codes and parity-check matrices for the dual codes can be determined. Several algebraic and geometric constructions of quasi-cyclic codes are presented as applications along with simulation results showing their performance over additive white Gaussian noise channels decoded with iterative message-passing algorithms.

Classification of (v, 3)-configurations

A (v, 3)-configuration is a nondegenerate matrix of dimension v over the field GF(2) considered up to permutation of rows and columns and containing exactly three 1’s in the rows and columns, while the inverse matrix has also exactly three 1’s in the rows and columns. It is proved that, for each even v ≥ 4, there is only one indecomposable (v, 3)-configuration, while, for odd v, there are no such configurations, the only exception being the unique (5, 3)-configuration.

THE COLUMN GROUP AND ITS LINK INVARIANTS

The column group is a subgroup of the symmetric group on the elements of a finite birack generated by the column permutations in the birack matrix. We use subgroups of the column group associated to birack homomorphisms to define an enhancement of the integral birack counting invariant and give examples which show that the enhanced invariant is stronger than the unenhanced invariant.

Evidence for a forbidden configuration conjecture: One more case solved

A simple matrix is a (0,1)-matrix with no repeated columns. Let F and A be (0,1)-matrices. We say that AavoidsF if there is no submatrix of A which is a row and column permutation of F. Let ‖A‖ denote the number of columns of A. We define forb(m,F)=max{‖A‖:A is an m-rowed simple matrix which avoids F}.For two matrices H and K, define [H∣K] as the concatenation of H and K. Let t⋅H denote the concatenation of t copies of H. Given a number t with t≥1, define F8(t)=[1010010111001100t⋅[10011100]].We are able to show that forb(m,F8(t)) is Θ(m2) and that this matrix is “maximal” (in some sense) with respect to this property. A conjecture of Anstee and Sali predicts three “maximal” 4-rowed cases to consider with quadratic bounds, and F8(t) is one of them. Establishing the quadratic upper bounds for all three cases would establish the veracity of the conjecture for all 4-rowed configurations.

Isomorphism examination based on the count vector

Isomorphism examination determines whether two design matrices are equivalent subject to some row, column, and level permutations. The purpose of this paper is to study the isomorphism problem from the viewpoint of the count vector. We find that two designs are isomorphic if and only if there exists a special type of linear transformation between their count vectors. The transformation can be characterized in terms of set operations for the subscripts of elements in the count vector. Besides, we propose an initial screening method based on the count vector, called the split-count matrix. We prove that the split-count matrix is more efficient than most existing initial screening methods. Some modified versions of the split-count matrix, including a projection version and some simplified versions, are discussed. Some examples and comparisons are given to demonstrate the power of the split-count matrix.

Genetic Algorithms Applied to Problems of Forbidden Configurations

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix $F$, we say a (0,1)-matrix $A$ avoids $F$ (as a configuration) if there is no submatrix of $A$ which is a row and column permutation of $F$. Let $\|{A}\|$ denote the number of columns of $A$. We define $\mathrm{forb}(m,F)=\max\{\|{A}\|\ : A$ is an $m$-rowed simple matrix that avoids $F \}$. Define an extremal matrix as an $m$-rowed simple matrix $A$ with that avoids $F$ and $\|{A}\|=\mathrm{forb}(m,F)$. We describe the use of Local Search Algorithms (in particular a Genetic Algorithm) for finding extremal matrices. We apply this technique to two forbidden configurations in turn, obtaining a guess for the structure of an $m\times\mathrm{forb}(m,F)$ simple matrix avoiding $F$ and then proving the guess is indeed correct. The Genetic Algorithm was also helpful in finding the proof.

Forbidden Configurations: Boundary Cases

Abstract A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix F, we say a (0,1)-matrix A has F ⊀ A , if no submatrix of A is a row and column permutation of F. Let the number of columns of A be ‖ A ‖ . Let forb ( m , F ) = max { ‖ A ‖ : A is m -rowed simple matrix and F ⊀ A } . A conjecture of Anstee and Sali predicts the asymptotics of forb ( m , F ) as a function of F. The conjecture identifies certain matrices F as boundary cases, namely k-rowed F with forb ( m , F ) = Θ ( m l ) while for any k × 1 column α that is not already present two or more times in F, we have forb ( m , [ F | α ] ) being Ω ( m l + 1 ) . We establish two new boundary cases and review two other newly identified boundary cases. This is further evidence for the conjecture.

Two-dimensional cache-oblivious sparse matrix–vector multiplication

In earlier work, we presented a one-dimensional cache-oblivious sparse matrix-vector (SpMV) multiplication scheme which has its roots in one-dimensional sparse matrix partitioning. Partitioning is often used in distributed-memory parallel computing for the SpMV multiplication, an important kernel in many applications. A logical extension is to move towards using a two-dimensional partitioning. In this paper, we present our research in this direction, extending the one-dimensional method for cache-oblivious SpMV multiplication to two dimensions, while still allowing only row and column permutations on the sparse input matrix. This extension requires a generalisation of the compressed row storage data structure to a block-based data structure, for which several variants are investigated. Experiments performed on three different architectures show further improvements of the two-dimensional method compared to the one-dimensional method, especially in those cases where the one-dimensional method already provided significant gains. The largest gain obtained by our new reordering is over a factor of 3 in SpMV speed, compared to the natural matrix ordering.

Forbidden configurations and repeated induction

For a given k×ℓ matrix F, we say a matrix A has no configurationF if no k×ℓ submatrix of A is a row and column permutation of F. We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. We define forb(m,F) as the maximum number of columns in an m-rowed simple matrix which has no configuration F. A fundamental result of Sauer, Perles and Shelah, and Vapnik and Chervonenkis determines forb(m,Kk) exactly, where Kk denotes the k×2k simple matrix. We extend this in several ways. For two matrices G,H on the same number of rows, let [G∣H] denote the concatenation of G and H. Our first two sets of results are exact bounds that find some matrices B,C where forb(m,[Kk∣B])=forb(m,Kk) and forb(m,[Kk∣Kk∣C])=forb(m,[Kk∣Kk]). Our final result provides asymptotic boundary cases; namely matrices F for which forb(m,F) is O(mp) yet for any choice of column α not in F, we have forb(m,[F∣α]) is Ω(mp+1). This is evidence for a conjecture of Anstee and Sali. The proof techniques in this paper are dominated by repeated use of the standard induction employed in forbidden configurations. Analysis of base cases tends to dominate the arguments. For a k-rowed (0,1)-matrix F, we also consider a function req(m,F) which is the minimum number of columns in an m-rowed simple matrix for which each k-set of rows contains F as a configuration.

Equivalence of factorial designs with qualitative and quantitative factors

Two symmetric fractional factorial designs with qualitative and quantitative factors are equivalent if the design matrix of one can be obtained from the design matrix of the other by row and column permutations, relabeling of the levels of the qualitative factors and reversal of the levels of the quantitative factors. In this paper, necessary and sufficient methods of determining equivalence of any two symmetric designs with both types of factors are given. An algorithm used to check equivalence or non-equivalence is evaluated. If two designs are equivalent the algorithm gives a set of permutations which map one design to the other. Fast screening methods for non-equivalence are considered. Extensions of results to asymmetric fractional factorial designs with qualitative and quantitative factors are discussed.

Small forbidden configurations V: Exact bounds for 4 × 2 cases

The present paper continues the work begun by Anstee, Ferguson, Griggs, Kamoosi and Sali on small forbidden configurations. We define a matrix to besimpleif it is a (0, 1)-matrix with no repeated columns. LetFbe ak× (0, 1)-matrix (the forbidden configuration). AssumeAis anm×nsimple matrix which has no submatrix which is a row and column permutation ofF. We define forb (m, F) as the largestn, which would depend onmandF, so that such anAexists.DefineFabcdas the (a+b+c+d) × 2 matrix consisting ofarows of [11],brows of [10],crows of [01] anddrows of [00]. With the exception ofF2110, we compute forb (m; Fabcd) for all 4 × 2Fabcd. A number of cases follow easily from previous results and general observations. A number follow by clever inductions based on a single column such as forb (m; F1111) = 4m− 4 and forb (m; F1210) = forb (m; F1201) = forb (m; F0310) = (2m)+m+ 2 (proofs are different). A different idea proves forb (m; F0220) = (2m) + 2m− 1 with the forbidden configuration being related to a result of Kleitman. Our results suggest that determining forb (m; F2110) is heavily related to designs and we offer some constructions of matrices avoidingF2110using existing designs.

Virtual Cylindrical View of a Color Image for its Permutation for an Encryption Purpose

This paper presents a novel algorithm for row and column permutation of pixels for the purpose of image encryption. The algorithm introduces a virtual cylinder surrounding an image and a virtual viewer looking at it but displaced from an original position. The key idea is based on the assumption that the light ray of each pixel to the viewer in her/his original position intersects the cylinder surface at a given point. When the viewer is displaced, the new position on a perpendicular image plane on which the pixel is projected should also have its direction intersecting the cylinder on the same point. As a result, all projected pixels in the new created image are slid from their original positions; but, some of them are delayed because they are piled up or projected out. In order to avoid information loss, these pixels are projected in the created holes of the new image. The consequence of such process is the creation of the expected permutation. Despite its simplicity, the algorithm shows a strong transformation of images for the purpose of their encryption.

Linear algebra methods for forbidden configurations

We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. Given m and a k×l (0,1)-matrix F we define forb(m,F) as the maximum number of columns in a simple m-rowed matrix A for which no k×l submatrix of A is a row and column permutation of F. In set theory notation, F is a forbidden trace. For all k-rowed F (simple or non-simple) Furedi has shown that forb(m,F) is O(m k ). We are able to determine for which k-rowed F we have that forb(m,F) is O(m k−1) and for which k-rowed F we have that forb(m,F) is Θ(m k ). We need a bound for a particular choice of F. Define D 12 to be the k×(2 k −2 k−2−1) (0,1)-matrix consisting of all nonzero columns on k rows that do not have [11] in rows 1 and 2. Let 0 denote the column of k 0’s. Define F k (t) to be the concatenation of 0 with t+1 copies of D 12. We are able to show that forb(m,F k (t)) is Θ(m k−1). Linear algebra methods and indicator polynomials originated in this context in a paper of the authors and Furedi and Sali. We provide a novel application of these methods. The results are further evidence for the conjecture of Anstee and Sali on the asymptotics for fixed F of forb(m,F).