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• W4386869661 abstract "Goppa, in the 1970s, discovered the relation between algebraic geometry and codes, which led to the family of Goppa codes. As one of the most interesting subclasses of linear codes, the family of Goppa codes is often chosen as a key in the McEliece cryptosystem. Knowledge of the number of inequivalent binary Goppa codes for fixed parameters may facilitate in the evaluation of the security of such a cryptosystem. Let <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n</i> ≥ 5 be an odd prime number, let <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>q</i> = 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><i>n</i></sup> and let <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>r</i> ≥ 3 be a positive integer satisfying gcd( <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>r, n</i> ) = 1. The purpose of this paper is to establish an upper bound on the number of inequivalent extended irreducible binary Goppa codes of length <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>q</i> + 1 and degree <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>r</i> . A potential mathematical object for this purpose is to count the number of orbits of the projective semi-linear group PGL <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sub> (F <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><i>q</i></sub> ) ⋊ Gal(F <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><i>q^r</i></sub> /F <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sub> ) on the set <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>I_r</i> of all monic irreducible polynomials of degree <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>r</i> over the finite field F <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><i>q</i></sub> . An explicit formula for the number of orbits of PGL <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sub> (F <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><i>q</i></sub> )⋊Gal(F <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><i>q^r</i></sub> /F <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sub> ) on <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>I_r</i> is given, and consequently, an upper bound for the number of inequivalent extended irreducible binary Goppa codes of length <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>q</i> +1 and degree <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>r</i> is derived. Our main result naturally contains the main results of Ryan (IEEE-TIT 2015), Huang and Yue (IEEE-TIT, 2022) and, Chen and Zhang (IEEE-TIT, 2022), which considered the cases <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>r</i> = 4, <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>r</i> = 6 and gcd( <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>r</i> , <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>q</i> <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>3</sup> - <italic xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>q</i> ) = 1 respectively." @default.
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• W4386869661 date "2023-01-01" @default.
• W4386869661 modified "2023-09-27" @default.
• W4386869661 title "The Number of Extended Irreducible Binary Goppa Codes" @default.
• W4386869661 doi "https://doi.org/10.1109/tit.2023.3317290" @default.
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