Warnaar's bijection and colored partition identities, I

We provide a general and unified combinatorial framework for a number of colored partition identities, which include the five, recently proved analytically by B. Berndt, that correspond to the exceptional modular equations of prime degree due to H. Schroeter, R. Russell and S. Ramanujan. Our approach generalizes that of S. Kim, who has given a bijective proof for two of these five identities, namely the ones modulo 7 (also known as the Farkas-Kra identity) and modulo 3. As a consequence of our method, we determine bijective proofs also for the two highly nontrivial identities modulo 5 and 11, thus leaving open combinatorially only the one modulo 23.Comment: Contains the first portion of the first author's MIT senior thesis (2011). Some minor revisions with respect to the previous version. To appear in JCT

Warnaar’s bijection and colored partition identities, II

In our previous paper (J. Comb. Theory Ser. A 120(1):28–38, 2013), we determined a unified combinatorial framework to look at a large number of colored partition identities, and studied the five identities corresponding to the exceptional modular equations of prime degree of the Schröter, Russell, and Ramanujan type. The goal of this paper is to use the master bijection of Sandon and Zanello (J. Comb. Theory Ser. A 120(1):28–38, 2013) to show combinatorially several new and highly nontrivial colored partition identities. We conclude by listing a number of further interesting identities of the same type as conjectures.Massachusetts Institute of Technology. Dept. of Mathematics

Linear Boolean classification, coding and "the critical problem"

The problem of constructing a minimal rank matrix over GF(2) whose kernel does not intersect a given set S is considered. In the case where S is a Hamming ball centered at 0, this is equivalent to finding linear codes of largest dimension. For a general set, this is an instance of "the critical problem" posed by Crapo and Rota in 1970. This work focuses on the case where S is an annulus. As opposed to balls, it is shown that an optimal kernel is composed not only of dense but also of sparse vectors, and the optimal mixture is identified in various cases. These findings corroborate a proposed conjecture that for annulus of inner and outer radius nq and np respectively, the optimal relative rank is given by (1-q)H(p/(1-q)), an extension of the Gilbert-Varshamov bound H(p) conjectured for Hamming balls of radius np

A proof that Reed-Muller codes achieve Shannon capacity on symmetric channels

Reed-Muller codes were introduced in 1954, with a simple explicit construction based on polynomial evaluations, and have long been conjectured to achieve Shannon capacity on symmetric channels. Major progress was made towards a proof over the last decades; using combinatorial weight enumerator bounds, a breakthrough on the erasure channel from sharp thresholds, hypercontractivity arguments, and polarization theory. Another major progress recently established that the bit error probability vanishes slowly below capacity. However, when channels allow for errors, the results of Bourgain-Kalai do not apply for converting a vanishing bit to a vanishing block error probability, neither do the known weight enumerator bounds. The conjecture that RM codes achieve Shannon capacity on symmetric channels, with high probability of recovering the codewords, has thus remained open. This paper closes the conjecture's proof. It uses a new recursive boosting framework, which aggregates the decoding of codeword restrictions on `subspace-sunflowers', handling their dependencies via an $L_p$ Boolean Fourier analysis, and using a list-decoding argument with a weight enumerator bound from Sberlo-Shpilka. The proof does not require a vanishing bit error probability for the base case, but only a non-trivial probability, obtained here for general symmetric codes. This gives in particular a shortened and tightened argument for the vanishing bit error probability result of Reeves-Pfister, and with prior works, it implies the strong wire-tap secrecy of RM codes on pure-state classical-quantum channels