Abstract
Let $n$ be an odd prime and $m>1$ be a positive integer. We produce an upper bound on the number of inequivalent extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$. Some examples are given to illustrate our results.
Highlights
This paper focuses on the class of codes called Goppa codes
We produce an upper bound on the number of inequivalent extended irreducible binary Goppa codes of degree 2m and length 2n + 1
As this paper is focused on irreducible Goppa codes we begin with the definition of irreducible Goppa codes
Summary
This paper focuses on the class of codes called Goppa codes Goppa codes are said to contain good parameters. This might be the reason why they are of high practical value. The McEliece cryptosystem is believed to be a cryptosystem which may have potential to withstand attack by quantum computers [3]. As this cryptosystem chooses a Goppa code at random as its key, knowledge of the number of inequivalent Goppa codes for fixed parameters may facilitate in the evaluation of the security of such a cryptosystem. The count employs the tools which were used to count the non-extended versions (see [8])
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