We show that permutation decoding can be used, finding PD groups, for some of the binary codes obtained from the adjacency matrix of the graphs at the vertices G, forn ~ 7. Special thanks are also addressed to the Ministry of Petroleum of the Republic of Angola ( scholarship), University of Natal (graduate assistantship) and South African National Research Foundation (NRF) (fellowship).
Introduction
By examining codes and designs derived from the primitive representations of the first two Janko groups, Key and Moori in [66, Section 7]. We also look at the issue of minimum weight generators of the codes and their duals.
Groups
Permutation groups
A permutation group is said to be 2-transitive (or doubly transitive) on D if it is transitive on the ordered pairs of different points of D. It is easy to show that the orbitals of G correspond one-to-one to the orbits on D of the stabilizer Ga.
Permutation representations
Most of the definitions of Section 2.1 apply to permutation representations by applying them to the permutation set that is the image of that representation. A permutation representation is said to be faithful if its core is the identity group, in which case G is isomorphic to the image of its permutation group, and we are back to the case of permutation groups.
G IGI la 1= IDI = IGal·
Primitive groups
If G is transitive on n, then all blocks of a G-invariant partition P have the same cardinality and G acts transitively on P. Also, G is imprimitive on D if G is transitive on D and G preserves some nontrivial partition D .
Rank-3 primitive permutation groups
- Symplectic groups
- Alternating groups
If U is completely isotropic, it is equivalent to any two vectors in U being orthogonal, which means that U ~ Ul. In the context of the projective space, the subspace U E P(V) is called completely isotropic, if O' = U, isotropic, if O' =1= 0, and nonisotropic if O' = 0.
Codes and Combinatorial Structures
Codes
Any linear code of length n over IFq contains a zero vector °E IFqn whose entries are all zero elements of the field. It follows that the minimum distance of the linear code C is the minimum weight of the code.
Designs
A design is trivial if every k set of points coincides with a block of the design. A counting argument proves the well-known relationship between the parameters (3.2). Especially. Given a labeling on the point and block sets of 1), the transpose of an incidence matrix of 1) is an incidence matrix of 1)t.
Graphs
The automorphism group Aut(r) of the graph r is the subgroup of Sv consisting of all automorphisms of r. The automorphism group of the triangular graph T(n) for n > 4 is the symmetric group Sn' This follows from a theorem of Whitney [94], which states that if r is a connected graph with more than 4 vertices, then Aut ( L(r)) = Aut(r).
Finite geometries
- Projective geometries
- Affine geometries
A hyperplane H of the projective geometry PGm-1(lFq ), in homogeneous coordinates is determined by the non-zero vector (YI,. The dimension of the geometry is the same as the dimension of the vector space and affine dimension of an element x +U in AG(V) is the dimension of U as a subspace of V The points of this space are all the vectors of V, the lines are the compositions of the I-dimensional subspaces of V, and the hyperplanes are the compositions of the (m-I)-dimensional subspaces of V .
Decoding schemes
- Nearest neighbour decoding
- Majority logic decoding
- Permutation decoding
If an error has occurred at the ith position, then there are at least ri - (t - 1) equations (check equations) of the systems Si = {y. In particular, explicit permutation decoding sets are found for the binary codes of the triangular graphs and the codes of graphs on triples.
Codes from Combinatorial Structures
Codes from designs
Knowing the number of vectors of each weight that exist in a code is essential to determine whether the supports of these vectors can form a design or not. Clearly, knowing the number of vectors of each weight that exist in a code is essential to determine whether the supports of these vectors can form a design.
Codes from graphs
C is spanned by the incidence vectors of the blocks of I. Let B be such a block and suppose that it has k vertices, and thus gives a vector of weight k in C. It is clear that C contains the entire incidence vector of every the set of k points, and so we see that C contains all vectors of weight 2 that have as nonzero inputs 1 and -1. It turns out that the code is also the l-(v,k, k) design code obtained by taking the rows of the adjacency matrix as the block incidence vectors; the automorphism set of this design will contain the automorphism set of the graph. In Chapter 9 we investigate the binary codes of these graphs and establish some of their properties.
Codes from geometries
Using this program with the necessary modifications, the triangular graph of T(n) and its code can be constructed. It has a minimum weight qr and the minimum weight vectors are multiples of the incidence vectors of r-fiats. Their minimum weight is generally unknown, and some new results are reported in [61, Section 7] that infer the minimum weight from the geometrical properties of the plans.
Codes from Groups
Introduction
Important information about these codes can be obtained from the theory of modular representations of groups. In particular, all binary codes derived from primitive permutation representations of these groups are studied. All of these codes are non-equivalent, except for the repetition code (IFJ) and its double, which appear in both representations of level 40.
Codes from primitive groups
Now notice that the adjacency matrix for the graph is simply the incidence matrix for the I-plan, so the I-plan is necessarily self-dual. The next two theorems deal with automorphism groups of designs and codes constructed from a finite primitive permutation group in the manner described in Theorem 5.2.1. Now the self-duality of V implies that ~ and Rj are vectors of weight k in C Now, since a permutes the coordinate positions of the k nonzero entries of ~ in Rj, we conclude that a is an automorphism of C.
Codes from the Janko groups J 1 and J 2
- The computations for J 1
- The computations for J 2
For each of the seven primitive representations, the permutation group was constructed and formed the trajectories of the stabilizer of a point. For each of the nontrivial orbits, the symmetric 1 design was formed as described in Theorem 5.2.1. Where J2 was the full group, there is another copy of the design for another track of the same length.
A conjecture of Key and Moori
Introduction
Although conjecture 6.1.1 is true for the Janko groups J1 and J2, and some other simple groups, we show here that this is not always true: we have found examples of finite simple groups G with a primitive representation that has a design D yields such that the automorphism group of G does not contain the automorphism group of D. Contrary to conjecture 6.1.1, for G = A6 of degree 15 we found two isomorphic designs such that the automorphism group of the design is neither the group Aut(A6). nor a true subgroup of Aut(A6) containing A6. Similarly, for G = Ag we found that the orbits of length 56 and 63 respectively for Ag of degree 120 yield designs with the property that the automorphism group is not Aut(Ag ) nor a real subgroup of Aut(Ag ) containing Ag . , if D is one of these designs, then Aut(D) is the orthogonal group 0;(2): 2 and Aut(Ag ) 1.
Symplectic groups
By Theorem 3.4.1, we have that the automorphism group of the design of points and planes, and thus also of its complementary design, is the full projective semi-linear group prL2m (q). The automorphism group of PSP2m(q) is discussed in Dieudonne [41, Chapter 4], but fully determined for the case where m = 2 and q are even, by Steinberg [89]: see also Carter [23] for a description . Essentially, the automorphism group is prSP2m(q) except when m = 2 and q > 2 are even, in which case it is this group extended by an involution a not in prL4(q).
Alternating groups
- Computations for A 6
- Computations for A g
Considering one of the representations of degree 15, an orbit of length 8 produces a design with automorphism group of order 20160. Our calculations show that the complete automorphism groups of the models are either Ag , 8g = Aut(Ag ) or the orthogonal group Ot (2): 2. Since the dimension of C is equal to the dimension of the hull (see Appendix A.2), it follows that C ~ Cl.
Binary Codes from Symplectic Groups
Introduction
When q is even, we still get the designs and codes, but the interesting binary codes in these cases are the design binary codes of the projective geometry of points and hyperplanes, with the largest semilinear projective set acting: they are wells. -well-known generalized Reed-Muller codes (see, for example, Theorem 6.2.1 or [5, Chapter 5]). Alternatively, this code can be obtained by taking the row space over IF'2 of a strongly regular graph adjacency matrix defined by the rank-3 operation of PSP2m(q). We derive the results through a series of lemmas in Section 7.2 and summarize our results in Theorem 7.2.11.
The binary codes
We will now prove a series of lemmas that lead to some properties of the codes C and Cl., where C in all lemmas is the binary code of the. We first show that a preserves subspaces and is therefore in the group of geometry. It is clear that each element of this group is an automorphism of the design, and so the result is proven.
Binary Codes of Triangular Graphs
Introduction
The code formed by the span of adjacency matrix is also the code of the 1-(n(n2-1), 2(n - 2), 2(n - 2)) design V obtained by taking the rows of adjacency matrix as the occurrence vectors of the blocks; the automorphism group in this design will contain the automorphism group in the graph, the latter of which is easily seen to be 5n. An alternative way to approach the designs, graphs and codes that we will look at is through the primitive rank-3 operation of the simple alternating group An, for n 2:: 5, on the 2-subsets (or duads) n{ 2} of a set n of size n.
The binary codes
Now the sum of the incidence vectors of the blocks defined by i disjoint duads will give a vector of C with weight 2i(n - 2i). In the case of even the minimum weight vectors are then the incidence vectors of the blocks of the design. It is clear that ifn is even, then C has a basis of minimum weight vectors since the incidence vectors of the blocks are the minimum weight vectors and spans C by definition.
Binary Codes from Graphs on Triples
Introduction
The binary codes
We introduce an ordering of the points and the spanning weight-4 vectors such that the generating matrix has the shape of an upper triangle. Proof: We order the points of P and a specific set of words u(6:.*) so that the generating matrix has the shape of an upper triangle. So we arrange the points of P and a specific set of words w(11") so that the generating matrix has the shape of an upper triangle.
Permutation Decoding
- Introduction
- Codes of the triangular graphs
- Codes of the graphs on triples
- PD-sets through computation
- Codes from A 6 and A g
- Binary codes of the Chang graphs
Using the fact that the group of the code is Sn, we can find PD sets for the code C. Proof: Arrange the points of the coordinate set P as described in Equations (10.3) and (10.4) so that the first n - 1 points are in the information positions. Note, however, that PD sets can also be easily found for this code using the sequence of points in Lemma 8.2.7, where the set S of n permutations given in Equation (10.5) will be a PD set for Cl form. The Gordon-bound is less than this number. ii) The permutations given in the set S must be written as permutations on the points g,P2 .. iii).
Programmes Al and A2
I Programme Al
PD-sets Through Computation
Minimum Weights
Gordon Bound
Construction of codes from graphs
Constructing the Chang codes
A PD-set for the Chang 2 code
Dembowski, Finite Geometries, Acquisition of Mathematics and Their General Properties, Band 44, Springer Publishing House, Berlin, Heidelberg, New York, 1968. Pless, Introduction to the Theory of Error Correcting Codes, Third ed., Wiley Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons Inc., New York, 1998. Rotman, An Introduction to the Theory of Groups, Fourth Edition, Springer-Verlag, New York, Inc., 1995.
