Codes from the Janko groups J 1 and J 2

Codes from Groups

5.3 Codes from the Janko groups J 1 and J 2

Theorem 5.2.6 If C is a linear code of length n of a symmetric 1 - (v, k, k) design V over a finite field IF'q, then the automorphism group of V is contained in the automorphism group of C.

Proof: Suppose that

V

is a

l-(v, k, k)

design with

P =

{Pl,P2,'" ,Pv} the point set ofV and B

=

{Bl , B2,'" ,Bv} the block set. Let A be an incidence matrix for V, thenP determines uniquely the rows ofA, since each point is incident with precisely k blocks. Ifa E Aut(V), then a sends Pi to Pj for 1:S i,j:S v and Bi, to ^Bjl where 1 :S i' ,^{j' :S} v, and a preserves the incidence relation. Now if C is a code from V, then we have that the columns ofA span C. Let ~ and Rj denote the i-th and j-th columns ofA respectively, with the entries of~ and R_j labelled as the blocks indices. Then ~ and R_j have each exactly k non-zero entries, since they represent the incidence relation of a point with the corresponding k blocks ofV. Now the self-duality ofV implies that ~ and R_j are weight k vectors in C Now since a permutes the coordinate positions of the k non-zero entries of ~ to R_{j ,} we deduce that a is an automorphism ofC. •

of J₂ by its outer automorphism, and shown that for some of the codes the same is true.

Note that J1 has no outer automorphisms, and thus is its own automorphism group, whereas J2 has an involutary outer automorphism, so its automorphism group, which we will denote by J2 , is a split extension of J2 by Z2, with double the order.

They have looked first at J1 , which is of order 175560, and its maximal subgroups and primitive permutation representations via the coset action on these subgroups, see [27, 44]. There are seven distinct primitive representations, of degree 266, 1045, 1463, 1540, 1596, 2926, and 4180, respectively. They have then looked at J2 , of order 604800, which has nine primitive representations, of degree 100, 280, 315, 525, 840, 1008, 1800, 2016 and 10080, respectively.

For each of these groups, using Magma [11], the designs and graphs as described in Theorem 5.2.1 were found, and found the p-rank of the designs for some small set of values of the primep. To aid in the classification, the dimension of the hull of the design for each of these primes were also found. Also looked for strongly regular graphs for each group, finding as a result three for J2•

5.3.1 The computations for J

₁

For each of the seven primitive representations, the permutation group were constructed and formed the orbits of the stabilizer of a point. For each of the non- trivial orbits, the symmetric 1-design as described in Theorem 5.2.1 was formed.

There are 245 designs formed in this manner and that none of them is isomorphic to any other of the designs in this set. In every case the full automorphism group of the design or graph is J_1. In all but 34 of the designs, the dimensions of the code or the hull over the set of primes given above distinguished the designs. For the 34 remaining, these occurred in 17 pairs in which the set of

dimensions for each pair was identical, but distinct from all the other pairs.

InTable 5.1, the first column gives the degree, the second the number of orbits, and the remaining columns give the length of the orbits of length greater than 1, with the number of that length in parenthesis behind the length in case there is more than one of that length. The pairs that had the same code dimensions

I

Degree

[it]

^length

^I D

266 4 132 110 12 11

1045 10 168(5) 56(3) 28 8

1463 21 120(7) 60(9) 20(2) 15(2) 12 1540 20 114(9) 57(6) 38(4) 19 1596 18 110(13) 55(2) 22(2) 11 2926 66 60(34) 30(27) 15(5) 4180 106 42(95) 21(6) 14(4) 7

Table 5.1: Orbits of the point-stabilizer of

J

1

occurred as follows: for degrees 266, 1045 and 1596, there were no such pairs; for degree 1463 there were two pairs, both for orbit size 60; for degree 1540, there were two pairs, for orbit size 57 and 114 respectively; for degree 2926 there was one pair for orbit size 60; for degree 4180 there were 12 pairs, for orbit size 42.

For each one of these 245 designs (or graphs) there was at least one prime from the small set that gave an "interesting code", that is a code that is not the full space or of codimension 1. Full details of the numbers obtained can be found at the web site:

http://www.ces.clemson.edu/-keyj/

under the file "Janko groups and designs".

Proposition 5.3.1 [66] IfG is the first Janko group J1 , there are precisely 245 non-isomorphic self-dual 1-designs obtained by taking all the images underG of the non-trivial orbits of the point stabilizer in any ofG's primitive representations, and on which G acts primitively on points and blocks. In each case the full automorphism group is J_{1 .} Every primitive action on symmetric i-designs can be obtained by taking the union of such orbits and orbiting underG.

5.3.2 The computations for J

₂

This group has nine primitive representations, as already mentioned, but computations were not carried with the largest degree. Thus the results cover only the first eight. The results for J₂ are different from those for J1 , due to the existence of an outer automorphism.

The main difference is that usually the full automorphism group of the design is J_{2 ,} and that in the cases where it was only J2 ,there would be another orbit of that length that would give an isomorphic design, and which, if two orbits were joined, would give a design of double the block size and automorphism group J_{2 .} A similar conclusion held ifsome union of orbits was taken as a base block.

From these eight primitive representations, in all 51 non-isomorphic symmetric designs on which J2 acts primitively were obtained. Table 5.2 gives the same information for J2 that Table 5.1 gives for J1 . .The automorphism group of the design in each case was J2or

J

2 . WhereJ2 was the full group, there is another copy of the design for another orbit of the same length. This occurred in the following cases: degree 315, orbit length 32; degree 1008, orbit lengths 60, 100 and 150;

degree 1800, orbit lengths 42, 42, 84 and 168; degree 2016, orbit lengths 50, 75, 75, 150, 150, and 300. We note again that the p-ranks of the designs and their hulls gave an initial indication of possible isomorphisms and clear non-isomorphisms, so that only the few mentioned needed be tested.

IT]

100 3 63 36

280 4 135 108 36

315 6 160 80 32(2) 10

525 6 192(2) 96 32 12

840 7 360 240 180 24 20 15

1008 11 300 150(2) 100(2) 60(2) 50 25 12 1800 18 336 168(6) 84(3) 42(3) 28 21 14(2) 2016 18 300(2) 150(6) 75(5) 50(2) 25 15

I

^Degree

lI!J

^length

^I

Table 5.2: Orbits of the point-stabilizer of J2

Three strongly regular graphs were found: that of degree 100 from the rank- 3 action, and two more of degree 280 from the orbits of length 135 and 36, giving strongly regular graphs with parameters (280,135,70,60) and (280,36,8,4) respectively. The full automorphism group is

J

2 in each case. Not all the representations were checked, but note that the representation of degree 280 is the only one with point stabilizer having exactly four orbits. Note that Bagchi [6]

found a strongly regular with parameters (280,144,44,80) admitting J_{2 .}

The computations carried out in [66] regarding the Janko groups J₁ and J2 have been used to conjecture that the automorphism groups of the designs obtained using the construction method outlined in Theorem 5.2.1, from a primitive representation of a simple group G will have the automorphism group Aut(G) as its automorphism group, unless the design is isomorphic to another one constructed in this way, in which case the automorphism group of the design will be a proper subgroup of the the Aut(G) containing G.

In [66], Key and Moori have not found a code that has automorphism group bigger than J1 or

J

2 but not equal to the full symmetric group.