Block diagonalization for algebra's associated with block codes

For a matrix *-algebra B, consider the matrix *-algebra A consisting of the symmetric tensors in the n-fold tensor product of B. Examples of such algebras in coding theory include the Bose-Mesner algebra and Terwilliger algebra of the (non)binary Hamming cube, and algebras arising in SDP-hierarchies for coding bounds using moment matrices. We give a computationally efficient block diagonalization of A in terms of a given block diagonalization of B, and work out some examples, including the Terwilliger algebra of the binary- and nonbinary Hamming cube. As a tool we use some basic facts about representations of the symmetric group.Comment: 16 page

On the Caratheodory rank of polymatroid bases

In this paper we prove that the Carath\'eodory rank of the set of bases of a (poly)matroid is upper bounded by the cardinality of the ground set.Comment: 7 page

Polyhedra with the Integer Caratheodory Property

A polyhedron P has the Integer Caratheodory Property if the following holds. For any positive integer k and any integer vector w in kP, there exist affinely independent integer vectors x_1,...,x_t in P and positive integers n_1,...,n_t such that n_1+...+n_t=k and w=n_1x_1+...+n_tx_t. In this paper we prove that if P is a (poly)matroid base polytope or if P is defined by a TU matrix, then P and projections of P satisfy the integer Caratheodory property.Comment: 12 page

Integer decomposition for polyhedra defined by nearly totally unimodular matrices

We call a matrix $A$ nearly totally unimodular if it can be obtained from a totally unimodular matrix $\tilde{A}$ by adding to each row of $\tilde{A}$ an integer multiple of some fixed row a^{\ssf T} of $\tilde{A}$. For an integer vector $b$ and a nearly totally unimodular matrix $A$, we denote by $P_{A,b}$ the integer hull of the set ${x\in{\Bbb R}^n\mid Ax\leq b}$. We show that $P_{A,b}$ has the integer decomposition property and that we can find a decomposition of a given integer vector $x\in kP_{A,b}$ in polynomial time

Semidefinite code bounds based on quadruple distances

Let $A(n,d)$ be the maximum number of $0,1$ words of length $n$, any two having Hamming distance at least $d$. We prove $A(20,8)=256$, which implies that the quadruply shortened Golay code is optimal. Moreover, we show $A(18,6)\leq 673$, $A(19,6)\leq 1237$, $A(20,6)\leq 2279$, $A(23,6)\leq 13674$, $A(19,8)\leq 135$, $A(25,8)\leq 5421$, $A(26,8)\leq 9275$, $A(21,10)\leq 47$, $A(22,10)\leq 84$, $A(24,10)\leq 268$, $A(25,10)\leq 466$, $A(26,10)\leq 836$, $A(27,10)\leq 1585$, $A(25,12)\leq 55$, and $A(26,12)\leq 96$. The method is based on the positive semidefiniteness of matrices derived from quadruples of words. This can be put as constraint in a semidefinite program, whose optimum value is an upper bound for $A(n,d)$. The order of the matrices involved is huge. However, the semidefinite program is highly symmetric, by which its feasible region can be restricted to the algebra of matrices invariant under this symmetry. By block diagonalizing this algebra, the order of the matrices will be reduced so as to make the program solvable with semidefinite programming software in the above range of values of $n$ and $d$.Comment: 15 page