Cumulative-Separable Codes

A t first Γ(L,G)-codes were introduced by V.D.Goppa [1] in 1970. These codes are a large and powerful class of error correcting codes. F.J. McWilliams and N.J. Sloane [2] defined these codes as the most important class of alternant codes. It is known that there are Γ(L,G)-codes that reach the Gilbert-Varshamov bound and that many Γ(L,G)-codes are placed in the Table of the best known codes [3]. It is noted also that Goppa codes are interesting for postquantum cryptography. There are four basic types of Γ(L,G)-codes: cyclic, separable, cumulative, and irreducible Goppa codes. In this paper we describe new type of Γ(L,G)-codes that we call cumulative-separable Goppa codes. We are motivated to study this class of Goppa codes, because, as it will be shown below, there are its subclasses that have improved estimations on minimum distance and dimension and that there exist codes of these subclasses that have parameters better than those for codes from the Table of the best known codes [3]. This paper is organized as follows. In Section II we review briefly the definitions that we will use in the paper. In Section III we describe subclasses of cumulative-separable Γ(L,G)-codes with improved estimations on the dimension and minimum distance. In Section IV the relations between codes from different subclasses of cumulative-separable codes are presented. In Sections V and VI theorems on estimations of the dimension and minimum distance of considered subclasses are presented.