MinT - Best Known (42, 42+37, s)-Nets in Base 16

Best Known (42, 42+37, s)-Nets in Base 16

(42, 42+37, 518)-Net over F₁₆ — Constructive and digital

Digital (42, 79, 518)-net over F₁₆, using

1 times m-reduction [i] based on digital (42, 80, 518)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 40, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(42, 42+37, 642)-Net over F₁₆ — Digital

Digital (42, 79, 642)-net over F₁₆, using

1 times m-reduction [i] based on digital (42, 80, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 40, 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]

(42, 42+37, 83146)-Net in Base 16 — Upper bound on s

There is no (42, 79, 83147)-net in base 16, because

1 times m-reduction [i] would yield (42, 78, 83147)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8345 494026 135084 939029 148419 450280 843181 238987 613379 538747 156295 523272 287496 970604 446877 192716 > 16⁷⁸ [i]