Research Joint Ventures and Optimal Emissions Taxation

This paper performs a comparison of two well known approaches for modelling R&D spillovers associated with investment in E-R&D, namely dAspremont-Jacquemin and Kamien-Muller-Zang. We show that there is little qualitative difference between the models in terms of total surplus delivered when selecting the optimal tax regime when there is precommitment under cooperative regimes in which firms coordinate expenditures to maximize joint profits. However, under non-cooperative regimes there is marked difference, with the model of Kamien- Muller-Zang leading to higher taxation rates when firms share information. Furthermore, we argue that the Kamien-Muller-Zang model is of questionable validity when modelling R&D on emissions reducing technology due to counter intuitive results showing a positive relationhip between R&D spillovers and emissions taxes.

Efficient Multi-Point Local Decoding of Reed-Muller Codes via Interleaved Codex

Reed-Muller codes are among the most important classes of locally correctable codes. Currently local decoding of Reed-Muller codes is based on decoding on lines or quadratic curves to recover one single coordinate. To recover multiple coordinates simultaneously, the naive way is to repeat the local decoding for recovery of a single coordinate. This decoding algorithm might be more expensive, i.e., require higher query complexity. In this paper, we focus on Reed-Muller codes with usual parameter regime, namely, the total degree of evaluation polynomials is $d=\Theta({q})$, where $q$ is the code alphabet size (in fact, $d$ can be as big as $q/4$ in our setting). By introducing a novel variation of codex, i.e., interleaved codex (the concept of codex has been used for arithmetic secret sharing \cite{C11,CCX12}), we are able to locally recover arbitrarily large number $k$ of coordinates of a Reed-Muller code simultaneously at the cost of querying $O(q^2k)$ coordinates. It turns out that our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that accessing $k$ locations is in fact cheaper than repeating the procedure for accessing a single location for $k$ times. Our estimation of success error probability is based on error probability bound for $t$-wise linearly independent variables given in \cite{BR94}

List decoding Reed-Muller codes over small fields

The list decoding problem for a code asks for the maximal radius up to which any ball of that radius contains only a constant number of codewords. The list decoding radius is not well understood even for well studied codes, like Reed-Solomon or Reed-Muller codes. Fix a finite field $\mathbb{F}$. The Reed-Muller code $\mathrm{RM}_{\mathbb{F}}(n,d)$ is defined by $n$-variate degree-$d$ polynomials over $\mathbb{F}$. In this work, we study the list decoding radius of Reed-Muller codes over a constant prime field $\mathbb{F}=\mathbb{F}_p$, constant degree $d$ and large $n$. We show that the list decoding radius is equal to the minimal distance of the code. That is, if we denote by $\delta(d)$ the normalized minimal distance of $\mathrm{RM}_{\mathbb{F}}(n,d)$, then the number of codewords in any ball of radius $\delta(d)-\varepsilon$ is bounded by $c=c(p,d,\varepsilon)$ independent of $n$. This resolves a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008], who among other results proved it in the special case of $\mathbb{F}=\mathbb{F}_2$; and extends the work of Gopalan [FOCS 2010] who proved the conjecture in the case of $d=2$. We also analyse the number of codewords in balls of radius exceeding the minimal distance of the code. For $e \leq d$, we show that the number of codewords of $\mathrm{RM}_{\mathbb{F}}(n,d)$ in a ball of radius $\delta(e) - \varepsilon$ is bounded by $\exp(c \cdot n^{d-e})$, where $c=c(p,d,\varepsilon)$ is independent of $n$. The dependence on $n$ is tight. This extends the work of Kaufman-Lovett-Porat [IEEE Inf. Theory 2012] who proved similar bounds over $\mathbb{F}_2$. The proof relies on several new ingredients: an extension of the Frieze-Kannan weak regularity to general function spaces, higher-order Fourier analysis, and an extension of the Schwartz-Zippel lemma to compositions of polynomials.Comment: fixed a bug in the proof of claim 5.6 (now lemma 5.5

Boston Hospitality Review: Summer 2013

Hospitality Management: Perspectives from Industry Advisors by Rachel Roginsky and Matthew Arrants -- Te Four ‘Ps’ of Hospitality Recruiting by John D. Murtha -- Te Morris Nathanson Design Collection by Christopher Muller -- Still Searching for Excellence by Bradford Hudso

Syndrome decoding of Reed-Muller codes and tensor decomposition over finite fields

Reed-Muller codes are some of the oldest and most widely studied error-correcting codes, of interest for both their algebraic structure as well as their many algorithmic properties. A recent beautiful result of Saptharishi, Shpilka and Volk showed that for binary Reed-Muller codes of length $n$ and distance $d = O(1)$, one can correct $\operatorname{polylog}(n)$ random errors in $\operatorname{poly}(n)$ time (which is well beyond the worst-case error tolerance of $O(1)$). In this paper, we consider the problem of `syndrome decoding' Reed-Muller codes from random errors. More specifically, given the $\operatorname{polylog}(n)$-bit long syndrome vector of a codeword corrupted in $\operatorname{polylog}(n)$ random coordinates, we would like to compute the locations of the codeword corruptions. This problem turns out to be equivalent to a basic question about computing tensor decomposition of random low-rank tensors over finite fields. Our main result is that syndrome decoding of Reed-Muller codes (and the equivalent tensor decomposition problem) can be solved efficiently, i.e., in $\operatorname{polylog}(n)$ time. We give two algorithms for this problem: 1. The first algorithm is a finite field variant of a classical algorithm for tensor decomposition over real numbers due to Jennrich. This also gives an alternate proof for the main result of Saptharishi et al. 2. The second algorithm is obtained by implementing the steps of the Berlekamp-Welch-style decoding algorithm of Saptharishi et al. in sublinear-time. The main new ingredient is an algorithm for solving certain kinds of systems of polynomial equations.Comment: 24 page