Alternant and Algebraic Geometric Codes

Abstract

Alternant codes are subfield subcodes of a generalised Reed–Solomon code over an extension field of ${\mathbb F}_q$. This is a large class of linear codes which includes BCH codes, one of the families of cyclic codes which appeared in Chapter 5. Although BCH codes are not asymptotically good, we will prove that there are asymptotically good alternant codes. Not only are alternant codes linear, and so easy to encode, they also have an algebraic structure which can be exploited in decoding algorithms. However, as with the codes constructed in Theorem 3.7, the construction of these asymptotically good alternant codes is probabilistic. We prove that such a code must exist without giving an explicit construction.