MinT - Best Known (98−65, 98, s)-Nets in Base 16

Best Known (98−65, 98, s)-Nets in Base 16

(98−65, 98, 65)-Net over F₁₆ — Constructive and digital

Digital (33, 98, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 98, 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]

(98−65, 98, 120)-Net in Base 16 — Constructive

(33, 98, 120)-net in base 16, using

12 times m-reduction [i] based on (33, 110, 120)-net in base 16, using
- base change [i] based on digital (11, 88, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(98−65, 98, 193)-Net over F₁₆ — Digital

Digital (33, 98, 193)-net over F₁₆, using

net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
  - a curve from the manYPoints database [i]

(98−65, 98, 3791)-Net in Base 16 — Upper bound on s

There is no (33, 98, 3792)-net in base 16, because

1 times m-reduction [i] would yield (33, 97, 3792)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 635 236945 902020 189939 272807 329228 590181 610600 869404 881600 028247 846832 561016 083596 490469 853576 512411 148845 171808 270586 > 16⁹⁷ [i]