Bernardo Gabriel Rodrigues

Permutation decoding for the binary codes from triangular graphs

By finding explicit PD-sets we show that permutation decoding can be used for the binary code obtained from an adjacency matrix of the triangular graph T(n) for any n ≥ 5.

Codes associated with triangular graphs and permutation decoding

Linear codes that can be obtained from designs associated with the complete graph on n vertices and its line graph, the triangular graph, are examined. The codes have length n choose 2, dimension n or n − 1, and minimum weight n − 1 or 2n − 4. The parameters of the codes and their automorphism groups for any odd prime are obtained and PD-sets inside the symmetric group S n are found for full permutation decoding for all primes and all integers n ≥ 6.

A self-orthogonal doubly even code invariant under M c L : 2

We examine a design D and a binary code C constructed from a primitive permutation representation of degree 275 of the sporadic simple group McL. We prove that Aut(C) = Aut(D) = McL : 2 and determine the weight distribution of the code and that of its dual. In Section 5, we show that for a word wi of weight i, where i ∈ {100, 112, 164, 176} the stabilizer (McL)wi is a maximal subgroup of McL. The words of weight 128 splits into three orbits C(128)1, C(128)2 and C(128)3, and similarly the words of weights 132 produces the orbits C(132)1 and C(132)2. For wi ∈ {C(128)1, C(128)2, C(132)1}, we prove that (McL : 2)wi is a maximal subgroup of McL. Further in Section 6, we deal with the stabilizers (McL : 2)wi by extending the results of Section 5 to McL : 2.

Ternary codes from graphs on triples

For a set @W of size n>=7 and @W^{^3^} the set of subsets of @W of size 3, we examine the ternary codes obtained from the adjacency matrix of each of the three graphs with vertex set @W^{^3^}, with adjacency defined by two vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively.

Codes from incidence matrices of graphs

We examine the p-ary codes, for any prime p, from the row span over

LCD codes from adjacency matrices of graphs

{\mathbb {F}_p}

Switched graphs of some strongly regular graphs related to the symplectic graph

of |V| × |E| incidence matrices of connected graphs Γ = (V, E), showing that certain properties of the codes can be directly derived from the parameters and properties of the graphs. Using the edge-connectivity of Γ (defined as the minimum number of edges whose removal renders Γ disconnected) we show that, subject to various conditions, the codes from such matrices for a wide range of classes of connected graphs have the property of having dimension |V| or |V| − 1, minimum weight the minimum degree δ(Γ), and the minimum words the scalar multiples of the rows of the incidence matrix of this weight. We also show that, in the k-regular case, there is a gap in the weight enumerator between k and 2k − 2 of the binary code, and also for the p-ary code, for any prime p, if Γ is bipartite. We examine also the implications for the binary codes from adjacency matrices of line graphs. Finally we show that the codes of many of these classes of graphs can be used for permutation decoding for full error correction with any information set.

Automorphism group and diameter of a graph

The paper studies quasi-symmetric 2-(64, 24, 46) designs supported by minimum weight codewords in the dual code of the binary code spanned by the lines of AG(3, 22). We classify up to isomorphism all designs invariant under automorphisms of odd prime order in the full automorphism group G of the code, being of order \(\vert G\vert = 2^{13} \cdot 3^{4} \cdot 5 \cdot 7\). We show that there is exactly one isomorphism class of designs invariant under an automorphisms of order 7, 15 isomorphism classes of designs with an automorphism of order 5, and no designs with an automorphism of order 3. Any design in the code that does not admit an automorphism of odd prime order has full group of order 2 m for some m ≤ 13, and there is exactly one isomorphism class of designs with full automorphism group of order 213.