Correcting Codes : Combinatorics , Algorithms and Applications ( Fall 2007 ) Lecture 7 : Family of Codes Sep 12 , 2007

For example, CH the family of Hamming code is a family of codes with ni = 2 − 1, ki = 2 − i − 1, di = 3 and R(CH) = 1, δ(CH) = 0. We will mostly work with family of codes from now on. This is necessary as we will study the asymptotic behavior of algorithms for codes, which does not make sense for a fixed code. For example, when we say we say that a decoding algorithm for a code C takes O(n) time, we would be implicitly assuming that C is a family of codes and that the algorithm has an O(n) running time when the block length is large enough. From now on, unless mentioned otherwise, whenever we talk about a code, we will be implicitly assuming that we are talking about a family of codes. Finally, note that the motivating question is to study the optimal tradeoff between R and δ.