MinT - Best Known (84−55, 84, s)-Nets in Base 16

Best Known (84−55, 84, s)-Nets in Base 16

(84−55, 84, 65)-Net over F₁₆ — Constructive and digital

Digital (29, 84, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 84, 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]

(84−55, 84, 120)-Net in Base 16 — Constructive

(29, 84, 120)-net in base 16, using

6 times m-reduction [i] based on (29, 90, 120)-net in base 16, using
- base change [i] based on digital (11, 72, 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]

(84−55, 84, 161)-Net over F₁₆ — Digital

Digital (29, 84, 161)-net over F₁₆, using

net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(84−55, 84, 3648)-Net in Base 16 — Upper bound on s

There is no (29, 84, 3649)-net in base 16, because

1 times m-reduction [i] would yield (29, 83, 3649)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8781 939334 989731 388044 733391 062564 835605 870500 111542 721284 738915 306557 964992 595403 416110 643776 364096 > 16⁸³ [i]