نشریه علوم ریاضی و انفورماتیک

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Mohammadi R. One-point Goppa Codes on Some Genus 3 Curves with Applications in Quantum Error-Correcting Codes. IJMSI 2021; 16 (1) :65-76
URL: http://ijmsi.ir/article-1-1284-fa.html

We investigate one-point algebraic geometric codes C_L(D, G) associated to maximal curves recently characterized by Tafazolian and Torres given by the affine equation y^l = f(x), where f(x) is a separable polynomial of degree r relatively prime to l. We mainly focus on the curve y⁴ = x³ +x and Picard curves given by the equations y³ = x⁴-x and y3 = x⁴ -1. As a result, we obtain exact value of minimum distance in several cases and get many records that don’t exist in MinT tables (tables of optimal parameters for linear codes), such as codes over F_7² of dimension less than 36. Moreover, using maximal Hermitian curves and their sub-covers, we obtain a necessary and sufficient condition for self-orthogonality and Hermitian self-orthogonally of C_L(D, G).