We consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems. In the deterministic setting, a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions, that require at least a quadratic number of subspaces. Moreover, we address reconstruction under the erasure of a subset of the norms; using the concepts of p -fusion frames and list decoding, we propose an algorithm that outputs a finite list of candidate signals, one of which is the correct one. In the random setting, we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program.

The maximum size of a binary code is studied as a function of its lengthn, minimum distanced, and minimum codeword weight \ssiw. This functionB(n,d,w) is first characterized in terms of its exponential growth rate in the limitn→∞ for fixed δ =d/nand ω =w/n. The exponential growth rate ofB(n,d,w) is shown to be equal to the exponential growth rate ofA(n,d) for 0 ≤ ω ≤ 1/2, and equal to the exponential growth rate ofA(n,d,w) for 1/2B(n,d,w) are derived using the semidefinite programming (SDP) method. These bounds yield a nonasymptotic improvement of the second Johnson bound and are tight for certain values of the parameters.

We survey recent generalizations and improvements of the linear programming method that involve semidefinite programming. A general framework using group representations and tools from graph theory is provided.

Upper bounds are derived for codes in Stiefel and Grassmann manifolds with given minimum chordal distance. They stem from upper bounds for codes in the product of unit spheres and projective spaces. The new bounds are asymptotically better than the previously known ones.

Upper bounds are derived for codes in the Grassmann manifold with given minimum chordal distance. They stem from upper bounds for codes in the product of unit spheres and projective spaces. The new bounds are asymptotically better than the previously known ones.

This paper develops a new method to obtain upper bounds for spherical codes, based on semidefinite programming. With this method we improve the previous bounds for the kissing number in several dimensions, as well as other classical problems like Tammes' problem.

In this paper, we develop the linear programming method to obtain bounds for the cardinality of Grassmannian codes endowed with the chordal distance. We obtain a bound and its asymptotic version that generalize the well-known bound for codes in the real projective space obtained by Kabatyanskiy and Levenshtein, and improve the Hamming bound for sufficiently large minimal distances

Type II Z/sub 4/-codes are introduced as self-dual codes over the integers modulo 4 containing the all-one vector and with Euclidean weights multiple of 8. Their weight enumerators are characterized by means of invariant theory. A notion of extremality for the Euclidean weight is introduced. Their binary images under the Gray map are formally self-dual with even weights. Extended quadratic residue Z/sub 4/-codes are the main example of this family of codes. They are obtained by Hensel lifting of the classical binary quadratic residue codes. Their binary images have good parameters. With every type II Z/sub 4/-code is associated via construction A modulo 4 an even unimodular lattice (type II lattice). In dimension 32, we construct two unimodular lattices of norm 4 with an automorphism of order 31. One of them is the Barnes-Wall lattice BW32.