A Gentle Introduction to Erasure Codes

It is intuitive that removing even a single point from a line disconnects it, but removing a finite set of points from a plane leaves it connected. A line disconnected by a single point. A plane remaining connected even with a few points removed. However, this basic fact leads to a non-trivial property of real and complex polynomials: not all...

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. --> .graph { display: block; width: 300px; height: 300px; margin: 0.5em 0.2em; } .graph-container { display: inline-block; vertical-align: top; max-width: 300px; } (This was discussed on r/math and Hacker News.) 1. Overview In this article, I hope to convince you that the quintic equation is unsolvable, in the sense that I can’t...

KaTeXMacros = { "\\iroot": "\\operatorname{iroot}", "\\Bits": "\\operatorname{Bits}", "\\Err": "\\operatorname{Err}", "\\NewtonRoot": "\\mathrm{N{\\small EWTON}\\text{-}I{\\small ROOT}}", }; 1. The algorithm Today I’m going to talk about the generalization of the integer square root algorithm to higher roots. That is, given \(n\) and...

(Note: this article is a summary of this thread on ompf2.) The usual method for sampling a sphere from a point outside the sphere is to calculate the angle of the cone of the visible portion and uniformly sample within that cone, as described in Shirley/Wang. However, one detail that is glossed over is that you still need to map from the sampled...