The minimum distance estimate is one of the central parameters for binary Goppa codes. Every binary Goppa code defined by a square-free polynomial of degree $ r $ has a minimum distance of at least $ 2r+1 $. In this paper, we develop a support-synthesis framework for binary Goppa codes that installs an independent Bose-Chaudhuri-Hocquenghem (BCH)-type distance certificate. A compression lemma encodes any $ s $ subfield parity checks as one extension-field check; choosing $ \alpha_i = \beta+b_i^{-1} $ realizes this check as a degree-one Goppa condition. For BCH checks, a Frobenius reduction leaves $ \nu $ representatives, yielding $ \Gamma_2(L, x-\beta) = B(n, \delta, \zeta) $. For $ g = (x-\beta)h $ square-free of degree $ r $, we obtain $ \Gamma_2(L, g) = B(n, \delta, \zeta)\cap \Gamma_2(L, h) $ and $ d(\cdot)\ge\max\{2r+1, \delta\} $. We give positive-dimension examples for all $ r\ge1 $ and $ \delta > 2r+1 $, and constant-rate families with a distance of $ n/\log n $ for fixed $ r $. BCH decoding transfers to the subcode. The result is a deliberate support-design mechanism, not a universal improvement of the classical bound.
Citation: Shuhua Liang, Guanghui Zhang. New lower bounds for the minimum distance of a class of binary Goppa codes[J]. AIMS Mathematics, 2026, 11(9): 30145-30161. doi: 10.3934/math.20261194
The minimum distance estimate is one of the central parameters for binary Goppa codes. Every binary Goppa code defined by a square-free polynomial of degree $ r $ has a minimum distance of at least $ 2r+1 $. In this paper, we develop a support-synthesis framework for binary Goppa codes that installs an independent Bose-Chaudhuri-Hocquenghem (BCH)-type distance certificate. A compression lemma encodes any $ s $ subfield parity checks as one extension-field check; choosing $ \alpha_i = \beta+b_i^{-1} $ realizes this check as a degree-one Goppa condition. For BCH checks, a Frobenius reduction leaves $ \nu $ representatives, yielding $ \Gamma_2(L, x-\beta) = B(n, \delta, \zeta) $. For $ g = (x-\beta)h $ square-free of degree $ r $, we obtain $ \Gamma_2(L, g) = B(n, \delta, \zeta)\cap \Gamma_2(L, h) $ and $ d(\cdot)\ge\max\{2r+1, \delta\} $. We give positive-dimension examples for all $ r\ge1 $ and $ \delta > 2r+1 $, and constant-rate families with a distance of $ n/\log n $ for fixed $ r $. BCH decoding transfers to the subcode. The result is a deliberate support-design mechanism, not a universal improvement of the classical bound.
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Shuhua Liang, Guanghui Zhang. New lower bounds for the minimum distance of a class of binary Goppa codes[J]. AIMS Mathematics, 2026, 11(9): 30145-30161. doi: 10.3934/math.20261194
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