MinT - Best Known (40, 75, s)-Nets in Base 16

Best Known (40, 75, s)-Nets in Base 16

(40, 75, 518)-Net over F₁₆ — Constructive and digital

Digital (40, 75, 518)-net over F₁₆, using

1 times m-reduction [i] based on digital (40, 76, 518)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 38, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(40, 75, 642)-Net over F₁₆ — Digital

Digital (40, 75, 642)-net over F₁₆, using

1 times m-reduction [i] based on digital (40, 76, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 38, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [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]

(40, 75, 83419)-Net in Base 16 — Upper bound on s

There is no (40, 75, 83420)-net in base 16, because

1 times m-reduction [i] would yield (40, 74, 83420)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 127316 388866 983372 583731 509865 573060 998394 575050 730091 666713 833017 633523 387961 414450 274351 > 16⁷⁴ [i]