MinT - Best Known (42, 77, s)-Nets in Base 16

Best Known (42, 77, s)-Nets in Base 16

(42, 77, 520)-Net over F₁₆ — Constructive and digital

Digital (42, 77, 520)-net over F₁₆, using

1 times m-reduction [i] based on digital (42, 78, 520)-net over F₁₆, using
- trace code for nets [i] based on digital (3, 39, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(42, 77, 642)-Net over F₁₆ — Digital

Digital (42, 77, 642)-net over F₁₆, using

3 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, 77, 115596)-Net in Base 16 — Upper bound on s

There is no (42, 77, 115597)-net in base 16, because

1 times m-reduction [i] would yield (42, 76, 115597)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 32 595508 642416 982802 855174 514866 865781 643442 233142 192894 404577 299084 313160 579289 822980 061536 > 16⁷⁶ [i]