Bordered Constructions of Self-Dual Codes from Group Rings and New Extremal Binary Self-Dual Codes

A counterexample to the first Zassenhaus conjecture

The structure of group algebras

We introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus conjecture for the group PSL(2, 19). We also prove the Zassenhaus conjecture for PSL(2, 23). In a second application we show that there are no normalized units of order 6 in the integral group rings of M 10 and PGL(2, 9). This completes the proof of a theorem of Kimmerle and Konovalov that shows that the prime graph question has an affirmative answer for all groups having an order divisible by at most three different primes.

http://www.groupsstandrews.org/2013/slides/Baechle.pdf

Rational Conjugacy of Torsion Units in Integral Group Rings of Non-Solvable Groups

In this article the study of the Prime Graph Question for the integral group ring of almost simple groups which have an order divisible by exactly 4 different primes is continued. We provide more details on the recently developed “lattice method” which involves the calculation of Littlewood-Richardson coefficients. We apply the method obtaining results complementary to those previously obtained using the HeLP-method. In particular the “lattice method” is applied to infinite series of groups for the first time. We also prove the Zassenhaus Conjecture for four more simple groups. Furthermore we show that the Prime Graph Question has a positive answer around the vertex 3 provided the Sylow 3-subgroup is of order 3.

On the Prime Graph Question for Integral Group Rings of 4-Primary Groups II

We describe G-codes, which are codes that are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a G-code is also a G-code. We give constructions of self-dual and formally self-dual codes in this setting and we improve the existing construction given in Hurley (Int J Pure Appl Math 31(3):319–335, 2006) by showing that one of the conditions given in the theorem is unnecessary and, moreover, it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative [72, 36, 16] Type II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48. We define quasi-G codes and give a construction of these codes.

/pdf/group-rings-g-codes-and-constructions-of-self-dual-and-hozrhhf0yx.pdf

Group rings, G -codes and constructions of self-dual and formally self-dual codes

Let denote the group ring of the group over the ring  Using an isomorphism between and a certain ring of matrices in conjunction with other techniques, the structure of the unit group of the group algebra of the dihedral group of order 8 over any finite field of chracteristic 2 is determined in terms of split extensions of cyclic groups

The Structure of the Unit Group of the Group Algebra ${\mathbb{F}}_{2^k}D_{8}$

Constructions for Self-Dual Codes Induced from Group Rings

We prove that the Zassenhaus conjecture is true for PSL(2,8) and PSL(2,17). This is a continuation of research initiated by Kimmerle, Hertweck and Hofert.

Joe Gildea

Papers

Group rings, G -codes and constructions of self-dual and formally self-dual codes

The Structure of the Unit Group of the Group Algebra ${\mathbb{F}}_{2^k}D_{8}$

Constructions for Self-Dual Codes Induced from Group Rings

Zassenhaus conjecture for integral group ring of simple linear groups

Bordered Constructions of Self-Dual Codes from Group Rings and New Extremal Binary Self-Dual Codes