Algebraic generalization of BCH-Goppa-Helgert codes

Based on the Mattsom-Solomon polynomial, a class of algebraic linear error-correcting codes is proposed, which includes the Bose-Chaudhuri-Hocquenghen (BCH) codes, Goppa codes, and Srivastava codes as subclasses. Several constructive bounds on the minimum distance of these codes are derived and are shown to be achievable using either Berlekamp's iterative decoding algorithm or Goppa's method based on divided difference. Moreover, it is shown that this class of codes asymptotically approaches the Varshamov-Gilbert bound as
n \rightarrow \infty
. Although some binary Goppa codes were previously known to have
n \leq 2^m, r \leq m \cdot t
, and
d \geq 2t+ 1
, it is shown that a much larger class of codes also possesses such parameters. Finally, shortened codes are considered. With a limited computer search, a number of good codes were found. It is also observed that the proposed codes have no fundamental difference from those recently given by Helgert.