Applications of the Projective Space PG (3,8) in Coding Theory

Siam Journal on Discrete Mathematics, 2003

We study certain projections of binary linear codes onto larger fields. These projections include the well-known projection of the extended Golay ] code onto the Hexacode over GF(4) and the projection of the Reed-Muller code R(2, 5) onto the unique self-dual code over GF(4). We give a characterization of these projections, and we construct several binary linear codes which have best known optimal parameters , [40,, [48, 21, 12], and [72, 31, 16] for instance. We also relate the automorphism group of a quaternary code to that of the corresponding binary code.

Gazi University Journal of Science, 2010

Projective geometry is used to decode and represent codes easily. Cameron [1] generated a binary linear code from PG(2,2). In this paper we construct a binary linear code from PG(3,2). Also we give a decoding rule for this code. A simulation study is given to compare this decoding algorithm with maximum likelihood decoding algorithm.

Journal of Combinatorial Theory, Series A, 2002

In this article, we determine the words of minimum weight in the code of the incidence system of s-versus t-flats in a finite projective space. Our proof depends on a few combinatorial results on the geometry of flats which may be of independent interest. We also give bounds for the minimum weight of the dual of this code and show that they are attained in many cases. The lower bound is a consequence of a general result on the dual code of an incidence system.

Electronic Notes in Discrete Mathematics, 2001

Consider the natural action of PGL 3 (q) on the projective plane PG 2 (q) over a finite field GF(q). In this paper we split a set of representatives of conjugacy classes of PGL 3 (q) into a disjoint union of subfamilies gathering the elements that have the same cycle type as a permutation of the points of PG 2 (q). Also, we count the number of elements in each subfamily. This allows us to obtain formulas for the number of orbits of the action of PGL 3 (q) on different sets of n-subsets and n-multisubsets of PG 2 (q). As an application we obtain explicit formulas for the number of isometry classes of different families of codes of dimension three.

Algebraic Coding Theory and Applications, 1979

Designs, Codes and Cryptography, 2008

We study codewords of small weight in the codes arising from Desarguesian projective planes. We first of all improve the results of K. Chouinard on codewords of small weight in the codes arising from PG(2, p), p prime. Chouinard characterized all the codewords up to weight 2 p in these codes. Using a particular basis for this code, described by Moorhouse, we characterize all the codewords of weight up to 2 p + ( p − 1)/2 if p ≥ 11. We then study the codes arising from PG(2, q = q 3 0 ). In particular, for q 0 = p prime, p ≥ 7, we prove that the codes have no codewords with weight in the interval [q + 2, 2q − 1]. Finally, for the codes of PG(2, q), q = p h , p prime, h ≥ 4, we present a discrete spectrum for the 123 26 V. Fack et al.

Designs, Codes and Cryptography, 2019

An outstanding folklore conjecture asserts that, for any prime p, up to isomorphism the projective plane P G(2, F p) over the field F p := Z/pZ is the unique projective plane of order p. Let π be any projective plane of order p. For any partial linear space X , define the inclusion number i(X , π) to be the number of isomorphic copies of X in π. In this paper we prove that if X has at most log 2 p lines, then i(X , π) can be written as an explicit rational linear combination (depending only on X and p) of the coefficients of the complete weight enumerator (c.w.e.) of the p-ary code of π. Thus, the c.w.e. of this code carries an enormous amount of structural information about π. In consequence, it is shown that if p > 2 9 = 512, and π has the same c.w.e. as P G(2, F p), then π must be isomorphic to P G(2, F p). Thus, the uniqueness conjecture can be approached via a thorough study of the possible c.w.e. of the codes of putative projective planes of prime order.

Asian Research Journal of Mathematics

Let (G, ∗) be a group and X any set, an action of a group G on X, denoted as G×X → X, (g, x) 7→ g.x, assigns to each element in G a transformation of X that is compatible with the group structure of G. If G has a subgroup H then there is a transitive group action of G on the set (G/H) of the right co-sets of H by right multiplication. A representation of a group G on a vector space V carries the dimension of the vector space. Now, given a field F and a finite group G, there is a bijective correspondence between the representations of G on the finitedimensional F-vector spaces and finitely generated FG-modules. We use the FG -modules to construct linear ternary codes and combinatorial designs from the permutation representations of the group L3(4). We investigate the properties and parameters of these codes and designs. We further obtain the lattice structures of the sub-modules and compare these ternary codes with the binary codes constructed from the same group.

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.

AL-Rafidain Journal of Computer Sciences and Mathematics, 2020

The purpose of this paper is to prove the existence of 17 new linear [337,