International Iran conference on Quantum Information September 2007, Kish Island

Evaluation of bounds of codes defined over hexagonal and honeycomb lattices via linear Evaluation of bounds of codes defined over hexagonal and honeycomb lattices via linear programming based on association schemesprogramming based on association schemes

M. A. Jafarizadeh, N. BehboodE-mail: n-behbood@hotmail.com

IntroductionWhile we can say much about codes without talking about association schemes, the algebraic structure they provide, gives us a more complete picture of how codes interact with the set they belong to. So far, Delsarte has used this algebraic structure to give one of the strongest bounds on the size of codes. Typically, the most of works on Delsarte's linear programming bound have been developed for the underlying codes of distance regular association schemes (distance-regular codes). In this work, by using underlying graphs of association schemes and Delsarte's linear programming bound, a way has been introduced for calculating some new bounds for more general codes which are not distance regular. Hence, for achieving upper bound for a given code with minimum distance d, the number of steps needed for covering all codewords in distance d from an arbitrary codeword (called reference codeword), is determined via the structure of the corresponding underlying graph of association scheme.

Association schemesLet X be a finite set of cardinality .Let . be subsets of such that:

Such a system is called an association scheme of class d on X. The cardinality of n on X is called the order of association scheme. The sets . are called the relations of the association scheme. In addition to the above properties: 5. If for all i, j, k is satisfied, then X is called a commutative association scheme of class d on X.6. If i’ = i for all i, then X is called a symmetric association scheme of class d on X.IdempotentSince the matrices Ai commute, they can be diagonalized simultaneously, i.e., there exists a matrix S such that for each is a diagonal matrix. Therefore A is semi-simple and has a second basis .These are matrices satisfying

The matrix (where J is the all-one matrix in A) is a minimal idempotent (idempotent is clear, and minimal follows from the rank(J)= 1). The are known as the primitive idempotent of Y . An idempotent is primitive if it is not the sum of two orthogonal idempotents.

Distribution vectors and Delsarte linear programming boundIf (X,R) is an association scheme and Y is a subset of X, then we can define the distribution vector and distribution matrix of Y as follows. The distribution vector of Y is the vector a whose ith entry is given by:

The `s can be thought of as the average number of elements in Y at distance i from some other element of Y. The linear programming bound, presented by Delsarte in 1973, improved the bounds of codes. we know if a is the distribution vector of a subset Y of an association scheme (X,R) with dual eigenmatrix Q, then aQ ≥ 0. It can be resulted from the above that for an association scheme A with dual eigenmatrix Q, diameter d, and distribution vector . Then any code C with minimum distance r in A satisfies:

When constructing a code C with a particular minimum distance, the above gives a restriction on the distribution vector of C. The first element of a will be 1, and if we want to restrict our bound to codes of minimum distance r, the entries through should all be zero and the remaining entries should be non-negative. So we wish to maximize the sum:

with bellow restriction:

The linear programming bound gives us this last relationship, and so the solution to this linear program will be an upper bound on the size of our code.

Construction of two-variable P-polynomial association schemes fromFirst we choose the ordering of elements of as follows

Where . We use the notation (k, l) for the element of the group. Clearly, (k, l)(k’, l’) = (k + k’, l + l’) and .Then the vertex set V of the graph will be .For the generating set

(all permutations of the simple roots and . together with the lowest root of the root lattice). Then, the orbits form a partition P. since any product of two orbits is invariant under symmetric group , the set of orbits is closed under multiplication. Also, if we use the notation , it can be shown that the intersection number:

Is independent of the choice of . .Therefore, the relation define an abelian association scheme. The corresponding adjacency matrices recursion would be as follow

Adjacency matrices of underlying graph for m =7.

Where and are at distance 2 from , and at distance 3 and and at distance 4 of reference codeword . So, the underlying graph of association scheme structure of this example is not distance regular graph and despite the distance regular once, distance between any two arbitrary codewords is not equal to the number of steps needed for going from one codeword to the another.

At following, we will use of Delsarte linear programming bound for finding upper bound of this code with minimum distance 3. For this aim, all the array of distribution vector of whole words which are in distance less than 3 must be vanished to zero. the words which belong to are in distance less than 3, so the related distribution vector array . are equal to zero in the objective function and constraints. For the following objective function with the related constraints by using linear programmingBound (Simplex method), the upper bound of code is achieved.

Construction of association scheme from two copies of hexagonal latticeWe extend the group by direct product with and obtain . as a vertex set for underlying graph of association scheme that we want to construct finite honeycomb lattice is equivalent to two copies of finite hexagonal lattice. adjacency matrix A corresponding to finite honeycomb lattice is

where

case m = 3The adjacency matrices of Bose-Mesner algebra are written as

The objective function and constraints of this case are as follow:

the upper bound of related code with minimum distance d ≤3 is |C|≤3.103.

Quantum codes over GF(4)One aspect for constructing quantum codes is codes on the Galois field GF(4). The elements of this field consist of with . One additive code C with length n on GF(4) is one additive subsets on

If one error- correction quantum code [n,k,d] exists, then there is answers for the following equations.

Some of investigated bound for codes over Goalies field GF(4) has been introduced on table III.

Table III

Conclusion: Delsarte linear programming method for achiving upper bound of codes under association scheme structure which are not distance regular has been studied. It is showed that the underlying graph of association schem stratifies the graph into d+1 disjoint union of strata ( associate class) which is different from stratification based on distance, distance, except for distance regular graphs.References [1]-P.Delsarte - An algebraic approach to the association schemes of coding theory -Philips Research Reports Supplements, (1973)

[2]-A.R.Calderbank , E.M.Rains , P.W.Shor , A.Sloane - Quantum error correction and orthogonal geometry -,[quant-ph/9605005] (1996)

[3]-H.Tarnanen - Upper bound on permutation codes via linear programming – Eroup . J .combinatorics(1999) 20(101-114)

The investigated upper bound for hexagonal lattice of the above example is |C| ≤7 and this code is perfect code:

The other investigated upper bound for hexagonal association scheme are introdused at table I.

Table I

Another association scheme structure on are the association scheme with the first adjacency matrix equal to:

The investigated upper bound for square lattice with the above first adjacency matrix are shown in table II.

The perfect code case has been introduced in follow: