MinT - Best Known (68−33, 68, s)-Nets in Base 16

Best Known (68−33, 68, s)-Nets in Base 16

(68−33, 68, 516)-Net over F₁₆ — Constructive and digital

Digital (35, 68, 516)-net over F₁₆, using

trace code for nets [i] based on digital (1, 34, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(68−33, 68, 578)-Net over F₁₆ — Digital

Digital (35, 68, 578)-net over F₁₆, using

trace code for nets [i] based on digital (1, 34, 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]

(68−33, 68, 49960)-Net in Base 16 — Upper bound on s

There is no (35, 68, 49961)-net in base 16, because

1 times m-reduction [i] would yield (35, 67, 49961)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 474 383778 867857 770317 006832 833716 694714 241122 948650 032838 841214 322079 182780 603266 > 16⁶⁷ [i]