Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

Abstract

We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on n input bits, each of which has approximate Fourier sparsity at most O(n 3 ) and randomized parity decision tree complexity Θ(n). This improves upon the recent work of Chattopadhyay, Mande and Sherif [5] both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).

Variety Evasive Subspace Families

“…They have applications to constructing explicit list-decodable codes with small list size [GX13, GWX16, KRZSW18, GRZ20] and explicit dimension expanders [FG15,GRX21]. Subspace designs were also used to prove lower bounds in communication complexity [CGS20].…”