MinT - Best Known (23, 23+57, s)-Nets in Base 16

Best Known (23, 23+57, s)-Nets in Base 16

(23, 23+57, 65)-Net over F₁₆ — Constructive and digital

Digital (23, 80, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 80, 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]

(23, 23+57, 98)-Net in Base 16 — Constructive

(23, 80, 98)-net in base 16, using

base change [i] based on digital (7, 64, 98)-net over F₃₂, using
- net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
    - a function field by SÃ©mirat [i]

(23, 23+57, 129)-Net over F₁₆ — Digital

Digital (23, 80, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 80, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(23, 23+57, 1865)-Net in Base 16 — Upper bound on s

There is no (23, 80, 1866)-net in base 16, because

1 times m-reduction [i] would yield (23, 79, 1866)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 135352 703741 947513 908063 413293 436719 939575 812369 140162 364992 079436 467575 921788 032725 769455 958896 > 16⁷⁹ [i]