The high-dimensional conference in Jerusalem is running with many exciting talks (and they are videotaped), and today in Tel Aviv there is a conference on Optimization and Discrete Geometry : Theory and Practice.
Today in Jerusalem, Leonard Schulman talked (video available!) about a recent breakthrough by Gil Cohen, Bernhard Haeupler, and Leonard Schulman in the recent paper Explicit Binary Tree Codes with Polylogarithmic Size Alphabet.
Tree codes
Tree codes are extremely important objects invented by Leonard Schulman in 1992-1996. You can read about it in Wigderson’s book Mathematics and Computation (see also this post) in a whole section about Error-correction of interactive communication, a theory pioneered by Leonard. Finding explicit good (linear distance, finite alphabet) tree codes is a very important problem.
(From the introduction of the new paper) “A tree code consists of a complete rooted binary tree (either infinite or of finite depth Image may be NSFW.
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) in which each edge is labeled by a symbol from an alphabet Image may be NSFW.
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. There is a natural one-to-one mapping assigning each binary string Image may be NSFW.
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to a path starting at the root, where Image may be NSFW.
Clik here to view. simply indicates which child is taken in each of the steps. For a tree code, such a path naturally maps to a string over the alphabet Image may be NSFW.
Clik here to view., which is formed by concatenating the symbols along the path. This way a tree code Image may be NSFW.
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encodes any binary string Image may be NSFW.
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over Image may be NSFW.
Clik here to view.. This encoding has an online characteristic because the encoding of any prefix does not depend on later symbols. In particular, any two distinct strings that agree in their first Image may be NSFW.
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symbols also have encodings that agree in their first Image may be NSFW.
Clik here to view. symbols. A tree code is said to achieve distance Image may be NSFW.
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if the encodings of any two strings differ in at least a Image may be NSFW.
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-fraction of the positions after their first disagreement. The rate of a tree code is Image may be NSFW.
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. A tree code is said to be asymptotically good if it achieves both constant distance Image may be NSFW.
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and a constant rate, namely, the alphabet size is Image may be NSFW.
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.”
The abstract:
“This paper makes progress on the problem of explicitly constructing a binary tree code with constant distance and constant alphabet size.
For every constant Image may be NSFW.
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1 we give an explicit binary tree code with distance Image may be NSFW.
Clik here to view. and alphabet size Image may be NSFW.
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, where n is the depth of the tree. This is the first improvement over a two-decade-old construction that has an exponentially larger alphabet of size Image may be NSFW.
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.
As part of the analysis, we prove a bound on the number of positive integer roots a real polynomial can have in terms of its sparsity with respect to the Newton basis – a result of independent interest.”
Problems on Exponential sums
A few years ago Cris Moore and Leonard Schulman proposed an explicit constructions for good tree codes in the paper Tree codes and a conjecture on exponential sums. The construction depends on still open conjectures on new types of exponential sums.
A paper by Leonard and me on polynomials with massive cancellations.
So let me use this opportunity to advertize a paper by Leonard and me Quasi-random multilinear polynomials which was just accepted for publication in the Israel Journal of Mathematics. Unexpected cancellation is always a very interesting phenomenon that we cherish and wish to understand. For example, we expect that the sum of the Mobius function for the first n integers cancels almost like that of a random walk. Depending on the meaning of “almost” this is the (known) prime number theorem, the (conjectured) Riemann hypothesis, or simply false, respectively.
A little less famous example (which I personally like) is the paper by Nikola Djokic an upper bound on the sum of signs of permutations with a condition on their prefix sets solving a problem by J. Feldman, H. Knorrer, and E. Trubowitz.
Our starting point is the determinant: It is a polynomial of degree Image may be NSFW.
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