Near-MDS codes from elliptic curves
Angela Aguglia, Luca Giuzzi, Angelo Sonnino
Abstract
Abstract We provide a geometric construction of $$[n,9,n-9]_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mn>9</mml:mn><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>9</mml:mn><mml:mo>]</mml:mo></mml:mrow><mml:mi>q</mml:mi></mml:msub></mml:math> near-MDS codes arising from elliptic curves with n $${\mathbb {F}}_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>F</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:math> -rational points. Furthermore, we show that in some cases these codes cannot be extended to longer near-MDS codes.
Topics & Concepts
MathematicsElliptic curveSchoof's algorithmSupersingular elliptic curveEdwards curveHessian form of an elliptic curveDivision polynomialsTwists of curvesCounting points on elliptic curvesDiscrete mathematicsTripling-oriented Doche–Icart–Kohel curveCryptographyPure mathematicsFamily of curvesCombinatoricsCode (set theory)Finite fieldLinear codeElliptic curve point multiplicationModular elliptic curveMinimum distanceBlock codeHamming codeSato–Tate conjectureCoding theoryCoding theory and cryptographyCryptography and Residue ArithmeticAdvanced Wireless Communication Techniques