For nonnegative integers q, n, d, let A q (n, d) denote the maximum cardinality of a code of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider the following upper bound on A q (n, d). For any k, let C k be the collection of codes of cardinality at most k. Then A q (n, d) is at most the maximum value of v∈[q] n x({v}), where x is a function C 4 → R + such that x(∅) = 1 and x(C) = 0 if C has minimum distance less than d, and such that the C 2 ×C 2 matrix (x(C ∪C )) C,C ∈C 2 is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in n. It yields the new upper bounds A 4 (6, 3) ≤ 176, A 4 (7, 3) ≤ 596, A 4 (7, 4) ≤ 155, A 5 (7, 4) ≤ 489, and A 5 (7, 5) ≤ 87.

For nonnegative integers n, d, w, let A(n, d, w) be the maximum size of a code C ⊆ F n 2 with constant weight w and minimum distance at least d. We consider two semidefinite programs based on quadruples of code words that yield several new upper bounds on A(n, d, w). The new upper bounds imply that A(22, 8, 10) = 616 and A(22, 8, 11) = 672. Lower bounds on A(22, 8, 10) and A (22,8,11) are obtained from the (n, d) = (22, 7) shortened Golay code of size 2048. It can be concluded that the shortened Golay code is a union of constant weight w codes of sizes A(22, 8, w).

We give an independent set of size 367 in the fifth strong product power of C 7 , where C 7 is the cycle on 7 vertices. This leads to an improved lower bound on the Shannon capacity of C 7 : Θ(C 7 ) ≥ 367 1/5 > 3.2578. The independent set is found by computer, using the fact that the set {t · (1, 7, 7 2 , 7 3 , 7 4 ) | t ∈ Z 382 } ⊆ Z 5 382 is independent in the fifth strong product power of the circular graph C 108,382 . Here the circular graph C k,n is the graph with vertex set Z n , the cyclic group of order n, in which two distinct vertices are adjacent if and only if their distance (mod n) is strictly less than k.

A set of k orthonormal bases of C d is called mutually unbiased if | e, f | 2 = 1/d whenever e and f are basis vectors in distinct bases. A natural question is for which pairs (d, k) there exist k mutually unbiased bases in dimension d. The (well-known) upper bound k ≤ d + 1 is attained when d is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain C * -algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of S d and S k .We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. Along the way we provide a novel explicit decomposition of a certain "L-shaped" permutation module into irreducible modules. We present numerical results for small d, k and low levels of the hierarchy. In particular, we obtain (numerical) sumof-squares proofs for the (well-known) fact that there do not exist d + 2 mutually unbiased bases in dimensions d = 2, 3, 4, 5.

No abstract