MinT - Best Known (67−31, 67, s)-Nets in Base 16

Best Known (67−31, 67, s)-Nets in Base 16

(67−31, 67, 518)-Net over F₁₆ — Constructive and digital

Digital (36, 67, 518)-net over F₁₆, using

1 times m-reduction [i] based on digital (36, 68, 518)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 34, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(67−31, 67, 642)-Net over F₁₆ — Digital

Digital (36, 67, 642)-net over F₁₆, using

1 times m-reduction [i] based on digital (36, 68, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 34, 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]

(67−31, 67, 85067)-Net in Base 16 — Upper bound on s

There is no (36, 67, 85068)-net in base 16, because

1 times m-reduction [i] would yield (36, 66, 85068)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 29 646620 786797 967134 902217 911508 594053 399870 365492 821354 398509 092857 210506 180176 > 16⁶⁶ [i]