MinT - Best Known (41−17, 41, s)-Nets in Base 16

Best Known (41−17, 41, s)-Nets in Base 16

Digital (24, 41, 520)-net over F₁₆, using

1 times m-reduction [i] based on digital (24, 42, 520)-net over F₁₆, using
- trace code for nets [i] based on digital (3, 21, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (24, 41, 660)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁴¹, 660, F₁₆, 17) (dual of [660, 619, 18]-code), using
- 15 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 11 times 0) [i] based on linear OA(16³⁸, 642, F₁₆, 17) (dual of [642, 604, 18]-code), using
  - trace code [i] based on linear OA(256¹⁹, 321, F₂₅₆, 17) (dual of [321, 302, 18]-code), using
    - extended algebraic-geometric code AG_e(F,303P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

There is no (24, 41, 263144)-net in base 16, because

1 times m-reduction [i] would yield (24, 40, 263144)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 461539 568934 449785 456785 157332 184278 112270 416156 > 16⁴⁰ [i]