Goppa codes

Goppa described a new class of linear noncyclic error-correcting codes in [1] and [2]. This paper is a summary of Goppa's work, which is not yet available in English. ^1 We prove the four most important properties of Goppa codes. 1) There exist q -ary Goppa codes with lengths and redundancies comparable to BCH codes. For the same redundancy, the Goppa code is typically one digit longer. 2) All Goppa codes have an algebraic decoding algorithm which will correct up to a certain number of errors, comparable to half the designed distance of BCH codes. 3) For binary Goppa codes, the algebraic decoding algorithm assumes a special form. 4) Unlike primitive BCH codes, which are known to have actual distances asymptotically equal to their designed distances, long Goppa codes have actual minimum distances much greater than twice the number of errors, which are guaranteed to be correctable by the algebraic decoding algorithm. In fact, long irreducible Goppa codes asymptotically meet the Gilbert bound.