Symmetric configurations for bipartite-graph codes

Designs, Codes and Cryptography, 2008

In this paper, we study the p-ary linear code C(PG(n,q)), q = p h , p prime, h ≥ 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n,q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n,q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n,q)) and we prove that finding the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2,q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar’s lower bound (Geom Dedicata 8:407–415, 1979).

There is a chain of polynomial codes that contains the simplex code of the projective plane over GF (q). It is related to Veroneseans of the plane. We show how to construct information sets for some of these codes using any dual hyperoval in such a plane. Also, the more general Veroneseans of hypersurfaces of degree i of projective space are considered and, related to this, a general transformation of codes and of sets of points in projective geometry that generalizes coding theoretic duality. We call it "duality of order i". If we ensure that a set of points is taken to another set of points in the same space then the transformation is invertible for generic sets of points. First order duality corresponds to the usual duality of codes and of matroids. Quadratic duality takes any 9 3 configurations to another 9 3 e.g. Pappus. One of the Steiner triple systems having 13 points is taken to the projective plane P G(2, 3) of order 3 using an abstract version of the third order duality. This leads to a construction of the triple system using 26 conics in P G(2, 3).

Advances in Mathematics of Communications, 2011

We examine the p-ary codes from incidence matrices of Paley graphs P (q) where q ≡ 1 (mod 4) is a prime power, and show that the codes are [ q(q−1) 4 , q − 1, q−1 2 ] 2 or [ q(q−1) 4 , q, q−1 2 ] p for p odd. By finding PD-sets we show that for q > 9 the p-ary codes, for any p, can be used for permutation decoding for full error-correction. The binary code from the line graph of P (q) is shown to be the same as the binary code from an incidence matrix for P (q).

Designs, Codes and Cryptography, 2014

The hulls of codes from the row span over F p , for any prime p, of incidence matrices of connected k-regular graphs are examined, and the dimension of the hull is given in terms of the dimension of the row span of A + k I over F p , where A is an adjacency matrix for the graph. If p = 2, for most classes of connected regular graphs with some further form of symmetry, it was shown by Dankelmann et al. (Des. Codes Cryptogr. 2012) that the hull is either {0} or has minimum weight at least 2k − 2. Here we show that if the graph is strongly regular with parameter set (n, k, λ, μ), then, unless k is even and μ is odd, the binary hull is non-trivial, of minimum weight generally greater than 2k − 2, and we construct words of low weight in the hull; if k is even and μ is odd, we show that the binary hull is zero. Further, if a graph is the line graph of a k-regular graph, k ≥ 3, that has an -cycle for some ≥ 3, the binary hull is shown to be non-trivial with minimum weight at most 2 (k − 2). Properties of the p-ary hulls are also established.

Discrete Mathematics, 2014

Linear codes arising from the row span over any prime field Fp of the incidence matrices of the odd graphs O k for k ≥ 2 are examined and all the main parameters obtained. A study of the hulls of these codes for p = 2 yielded that for O2 (the Petersen graph), the dual of the binary hull from an incidence matrix is the binary code from points and lines of the projective geometry P G3(F2), which leads to a correspondence between the edges and vertices of O2 with the points and a collection of ten lines of P G3(F2), consistent with the codes. The study also gives the dimension, the minimum weight, and the nature of the minimum words, of the binary codes from adjacency matrices of the line graphs L(O k).

AL-Rafidain Journal of Computer Sciences and Mathematics, 2020

2007

In this paper, we study the p-ary linear code C(PG(n,q)), q = ph, p prime, h 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n,q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n,q) and we exclude all possible

ArXiv, 2020

In this paper, we examine the binary linear codes with respect to Hamming metric from incidence matrix of a unit graph $G(\mathbb{Z}_{n})$ with vertex set is $\mathbb{Z}_{n}$ and two distinct vertices $x$ and $y$ being adjacent if and only if $x+y$ is unit. The main parameters of the codes are given.

Finite Fields and Their Applications, 2018

Consider the Grassmann graph formed by k-dimensional subspaces of an n-dimensional vector space over the field of q elements (1 < k < n − 1) and denote by Π(n, k)q the restriction of this graph to the set of projective [n, k]q codes. In the case when q ≥ n 2 , we show that the graph Π(n, k)q is connected, its diameter is equal to the diameter of the Grassmann graph and the distance between any two vertices coincides with the distance between these vertices in the Grassmann graph. Also, we give some observations concerning the graphs of simplex codes. For example, binary simplex codes of dimension 3 are precisely maximal singular subspaces of a non-degenerate quadratic form.

The postgraduate scholarship I got from German Academic Exchange Service (Deutscher Akademischer Austausch Dienst-DAAD). To this end, I acknowledge the role played by the African Mathematical Sciences Institute through Gudrun Schirge and Liesl Jones in facilitating the scholarship.

Discrete Mathematics, 2011

We examine the p-ary linear codes from incidence matrices of the three uniform subset graphs with vertex set the set of subsets of size 3 of a set of size n, with adjacency defined by two vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively. All the main parameters of the codes and the nature of the minimum words are obtained, and it is shown that the codes can be used for full error-correction by permutation decoding. We examine also the binary codes of the line graphs of these graphs.

Discrete Mathematics, 1977

IEEE Transactions on Information Theory, 1999

New constructions of linear nonbinary codes with covering radius R = 2 are proposed. They are in part modifications of earlier constructions by the author and in part are new. Using a starting code with R = 2 as a "seed" these constructions yield an infinite family of codes with the same covering radius. New infinite families of codes with R = 2 are obtained for all alphabets of size q 4 and all codimensions r 3 with the help of the constructions described. The parameters obtained are better than those of known codes. New estimates for some partition parameters in earlier known constructions are used to design new code families. Complete caps and other saturated sets of points in projective geometry are applied as starting codes. A table of new upper bounds on the length function for q = 4; 5; 7; R = 2; and r 24 is included.

IEEE Transactions on Information Theory, 2005

New algebraic methods for constructing codes based on hyperplanes of two different dimensions in finite geometries are presented. The new construction methods result in a class of multistep majority-logic decodable codes and three classes of low-density parity-check (LDPC) codes. Decoding methods for the class of majority-logic decodable codes, and a class of codes that perform well with iterative decoding in spite of having many cycles of length 4 in their Tanner graphs, are presented. Most of the codes constructed can be either put in cyclic or quasi-cyclic form and hence their encoding can be implemented with linear shift registers.

IEEE Transactions on Information Theory, 1972