On True Minimum Distances of Goppa Codes

Abstract

Binary Goppa codes are important in coding theory and cryptography. It is well-known that the minimum distance of a binary Goppa code defined by a squarefree polynomial of degree r is at least 2r+1. Though from the classical results of C. J. Moreno, O. Moreno, F. Levy-dit-Vehel and S. Litsyn that the true minimum distances of binary Goppa codes are 2r+1, when the extension field degree m is larger than (r−1)^4r. There have only very few families of binary Goppa codes with determined true minimum distances. In this paper, we give necessary and sufficient conditions for binary Goppa codes such that their minimum distances are 2r+1 and 2r+2. Infinitely many infinite families of binary Goppa codes with minimum distance 2r+1 are given. In particular, we construct such codes for every Goppa degree r in the range 2^h−1<r<2^h over 𝔽_2^2h. For every 2≤ r≤⌊2^m/3⌋, we also obtain at least 2^m−3r+1 distinct monic squarefree Goppa polynomials of degree r over 𝔽_2^m, such that their related binary Goppa codes have minimum distance 2r+1. We extend a criterion for q-ary Goppa codes to split auxiliary polynomials and obtain some infinite families of q-ary Goppa codes attaining their designed distances r+1. We also construct infinite families of binary Goppa codes with minimum distance exactly 2r+2.

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BibTeX RIS

BibTeX

@misc{chen2026true,
  title = {{On True Minimum Distances of Goppa Codes}},
  author = {Yaqi Chen and Hao Chen and Chao Liu and Huimin Lao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26450},
  eprint = {2610.26450},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26450},
}

RIS

TY  - UNPB
TI  - On True Minimum Distances of Goppa Codes
AU  - Yaqi Chen
AU  - Hao Chen
AU  - Chao Liu
AU  - Huimin Lao
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26450
UR  - https://arxiv.org/abs/2610.26450
AB  - Binary Goppa codes are important in coding theory and cryptography. It is well-known that the minimum distance of a binary Goppa code defined by a squarefree polynomial of degree r is at least 2r+1. Though from the classical results of C. J. Moreno, O. Moreno, F. Levy-dit-Vehel and S. Litsyn that the true minimum distances of binary Goppa codes are 2r+1, when the extension field degree m is larger than (r−1)^4r. There have only very few families of binary Goppa codes with determined true minimum distances. In this paper, we give necessary and sufficient conditions for binary Goppa codes such that their minimum distances are 2r+1 and 2r+2. Infinitely many infinite families of binary Goppa codes with minimum distance 2r+1 are given. In particular, we construct such codes for every Goppa degree r in the range 2^h−1<r<2^h over 𝔽_2^2h. For every 2≤ r≤⌊2^m/3⌋, we also obtain at least 2^m−3r+1 distinct monic squarefree Goppa polynomials of degree r over 𝔽_2^m, such that their related binary Goppa codes have minimum distance 2r+1. We extend a criterion for q-ary Goppa codes to split auxiliary polynomials and obtain some infinite families of q-ary Goppa codes attaining their designed distances r+1. We also construct infinite families of binary Goppa codes with minimum distance exactly 2r+2.
ER  -

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