Optimal frames for erasures

Abstract

We study frames from the viewpoint of coding theory. We introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting.

The Kadison–Singer Problem in mathematics and engineering

“…Formally, this is a max͞min problem, which looks like: This problem has proved to be untouchable at this time. We only have a complete solution to the problem for two erasures (25)(26)(27). It was hoped that some special cases of the problem would be more tractible and serve as a starting point for the classification because the frames we are looking for are contained in this class.…”

On reflectionless equi-transmitting matrices

“…Five-and six-parameter families of 6 × 6 RET-matrices Conclusion 5.5. After our work was accomplished Ingemar Bengtsson pointed out, that similar matrices appear in the coding theory [7]. In fact 4-dimensional RET-matrices (4.4) can be found there as well as the 6-dimensional matrix D l 1 .…”

Generalized Regular Sampling of Trigonometric Polynomials and Optimal Sensor Arrangement

“…Corollary 2: is an optimal sensor arrangement if and only if it leads to a TF, in which case (11) While there are several mechanisms for findings sets of tight frames [9]- [11], the difficulty of our problem arises from the constraint that the frame vectors have forms fixed by the sampling of a trigonometric polynomial (1) (or its equivalent for higher dimensions). No full characterization of such tight frames is known; we provide novel sufficient conditions.…”

“…Note in particular that these fall outside all equivalences defined in [9], [11]. We refer to them as the generalized regular arrangements.…”