Jamshid Moori

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.

The Fischer-Clifford matrices of a maximal subgroup of ′₂₄

The Fischer group F i24 is the largest sporadic simple Fischer group of order 1255205709190661721292800 = 2.3.5.7.11.13.17.23.29 . The group F i24 is the derived subgroup of the Fischer 3-transposition group F i24 discovered by Bernd Fischer. There are five classes of elements of order 3 in F i24 as represented in ATLAS by 3A, 3B, 3C, 3D and 3E. A subgroup of F i24 of order 3 is called of type 3X, where X ∈ {A,B, C,D,E}, if it is generated by an element in the class 3X. There are six classes of maximal 3-local subgroups of F i24 as determined by Wilson. In this paper we determine the Fischer-Clifford matrices and conjugacy classes of one of these maximal 3-local subgroups Ḡ := Fi24 (〈N〉) ∼= 3·O7(3), where N ∼= 37 is the natural orthogonal module for Ḡ/N ∼= O7(3) with 364 subgroups of type 3B corresponding to the totally isotropic points. The group Ḡ is a nonsplit extension of N by G ∼= O7(3).

ON THE SPREAD OF THE SPORADIC SIMPLE GROUPS

A group is 2-generated if it can be generated by two elements x and y. In this case y is called a mate for x. Brenner and Wiegold defined a finite group G to have spread r. A group is said to have exact spread r if it has spread r but not r + 1. The exact spread of a group G is denoted by s(G). Ganief [11] in his PhD thesis proved that if G is a sporadic simple group, then s(G) ≥ 2. In this paper, by using probabilistic methods, for each sporadic simple group G we find a reasonable lower bound for s(G). *The second author was supported by research grants from URF (University of Natal) and NRF (South Africa).

(2,3, t)-Generations for the janko group j3

A group G is (l,m,n)-generated if it is a quotient group of the triangle group T(l,m,n) = (x,y,z|x l= y m= z n= xyz= 1). In [8] the problem is posed to find all possible (l,m,n)-generations for the non-abelian finite simple groups. In this paper we partially answer this question for the Janko group J 3. We find all (2, 3, t)-generations as well as (2, 2,2,p)-generations, p a prime, for J 3

SUBGROUPS OF 3-TRANSPOSITION GROUPS GENERATED BY FOUR 3-TRANSPOSITIONS

Abstract In this paper we study the subgroups of 3-transposition groups generated by a set of four 3-transpositions. As a consequence we deduce that the smallest Fischer simple group F 22 cannot be generated by four 3-transpositions and hence rank (F 22: D) = 5 or 6, where D is the conjugacy class of 3-transpositions in F 22 and rank (F 22: D) denotes the minimum number of 3-transpositions in D generating F 22. This result is an extension of the work done by the author in [11].

On the Exact Spread of Sporadic Simple Groups

A group is 2-generated if it can be generated by two elements x and y. In this case y is called a mate for x. Brenner and Wiegold (1975a) defined a finite group G to have spread r if for every set {x 1, x 2,…, x r } of distinct nontrivial elements of G, there exists an element y ∈ G such that G = ⟨ x i , y⟩ for all i. A group is said to have exact spread r if it has spread r but not r + 1. The exact spread of a group G is denoted by s(G). Ganief (1996) in his Ph.D. thesis proved that if G is a sporadic simple group, then s(G) ≥ 2. In Ganief and Moori (2001) the second author and Ganief used probabilistic methods and established a reasonable lower bound for the exact spread s(G) of each sporadic simple group G. The present article deals with the search for reasonable upper bounds for the exact spread of the sporadic simple groups.

rX-Complementary generations of the janko groups j1 J2 And j3

A finite group G with conjugacy class rX is said to be rX-complementary generated if, given an arbitrary x∈G-1, there is a y ∈ rXsuch that G (x,y). The rX-complementary generation of the simple groups was first introduced by Woldar in [17] to show that every sporadic simple group can be generated by an arbitrary element and another suitable element. It is conjectured in [5] that every finite simple group can be generated in this way. In this paper we investigate the rX-complementary generation of the first three Janko groups in an attemp to further develop the techniques of finding rX-complementary generation of the finite simple groups. As a consequence, we obtained all the(p,q,r)-generations of the Janko group J 3, where p,q,r are distinct primes.