MinT - Best Known (96−57, 96, s)-Nets in Base 16

Best Known (96−57, 96, s)-Nets in Base 16

(96−57, 96, 114)-Net over F₁₆ — Constructive and digital

Digital (39, 96, 114)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 33, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]
2. digital (6, 63, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(96−57, 96, 177)-Net in Base 16 — Constructive

(39, 96, 177)-net in base 16, using

base change [i] based on digital (7, 64, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(96−57, 96, 208)-Net over F₁₆ — Digital

Digital (39, 96, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 96, 208)-net over F₁₆, using
- net from sequence [i] based on digital (37, 207)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 37 and N(F) ≥ 208, using
    - Drinfeld modules of rank 1 [i]

(96−57, 96, 9153)-Net in Base 16 — Upper bound on s

There is no (39, 96, 9154)-net in base 16, because

1 times m-reduction [i] would yield (39, 95, 9154)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 465513 255684 669063 045069 426975 000548 723519 661572 582596 309498 092285 520640 134774 412168 203334 665915 627037 876127 987056 > 16⁹⁵ [i]