Lecture 4: Codes based on Concatenation

In the last lecture, we studied codes based on univarite polynomials, also known as Reed-Solomon codes, and on multivariate polynomials, also known as Reed-Muller codes. The Reed-Solomon codes had rate R = 1 − δ, which is optimal (given by the Singleton bound), but required alphabets of large size (|Σ| ≥ n). Multivariate polynomials in a constant number of variables, however, gave codes of positive rate and distance, and required an alphabet size of only ∼ n 1 m , where m is the number of variables. While this is not as bad as the Reed-Solomon codes in this aspect, it still is a growing function of n.