MinT - Best Known (78−35, 78, s)-Nets in Base 16

Best Known (78−35, 78, s)-Nets in Base 16

(78−35, 78, 522)-Net over F₁₆ — Constructive and digital

Digital (43, 78, 522)-net over F₁₆, using

trace code for nets [i] based on digital (4, 39, 261)-net over F₂₅₆, using
- net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(78−35, 78, 642)-Net over F₁₆ — Digital

Digital (43, 78, 642)-net over F₁₆, using

4 times m-reduction [i] based on digital (43, 82, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 41, 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]

(78−35, 78, 136075)-Net in Base 16 — Upper bound on s

There is no (43, 78, 136076)-net in base 16, because

1 times m-reduction [i] would yield (43, 77, 136076)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 521 512225 318703 396793 864590 051248 479831 757366 198869 057796 770328 525217 159453 032289 920744 476856 > 16⁷⁷ [i]