MinT - Best Known (117−57, 117, s)-Nets in Base 16

Best Known (117−57, 117, s)-Nets in Base 16

(117−57, 117, 516)-Net over F₁₆ — Constructive and digital

Digital (60, 117, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (60, 118, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 59, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(117−57, 117, 578)-Net over F₁₆ — Digital

Digital (60, 117, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (60, 118, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 59, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(117−57, 117, 73336)-Net in Base 16 — Upper bound on s

There is no (60, 117, 73337)-net in base 16, because

1 times m-reduction [i] would yield (60, 116, 73337)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 47 644190 733557 564503 699558 992598 050523 019588 626986 511767 911860 070077 514920 778013 401092 111322 279712 080455 493466 171244 727622 175289 866737 535816 > 16¹¹⁶ [i]