Chain of Separable Binary Goppa Codes and their Minimal Distance

TL;DR

This paper demonstrates that certain subclasses of separable binary Goppa codes can be organized into a chain of embedded codes and determines their exact minimal distances.

Contribution

It introduces a new chain structure for subclasses of separable binary Goppa codes and calculates their true minimal distances.

Findings

Codes form a chain of embedded codes

Exact minimal distances are determined for all codes in the chain

Provides new insights into the structure and parameters of Goppa codes

Abstract

It is shown that subclasses of separable binary Goppa codes, $\Gamma(L,G)$ - codes, with $L=\{\alpha \in GF(2^{2l}):G(\alpha)\neq 0 \}$ and special Goppa polynomials G(x) can be presented as a chain of embedded codes. The true minimal distance has been obtained for all codes of the chain.