Dense Sphere Packings from Perfect Codes -- Katherine Crowley, December 4, 2001

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In this talk we will discuss the basic ideas of coding theory and sphere 
packing, and the relationship between them. A good code has short codewords 
(for fast transmission of messages), many different codewords (to transmit a 
wide variety of information), and codewords that bear little resemblance to each 
other (in order to correct many errors). The goal of coding theory is to 
balance the conflicting aims of sending a message efficiently, and being 
able to correct errors that may occur when the codewords are transmitted over a 
noisy channel. The goal of sphere packing is to fit as many balls of the 
same size as densely as possible into a given space. We will see how 
"perfect" codes (codes that are both efficient and good at correcting errors) 
lead to dense sphere packings in <B>R</B>^n for certain dimensions n.