MinT - Best Known (58−35, 58, s)-Nets in Base 16

Best Known (58−35, 58, s)-Nets in Base 16

(58−35, 58, 82)-Net over F₁₆ — Constructive and digital

Digital (23, 58, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 17, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (6, 41, 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]

(58−35, 58, 120)-Net in Base 16 — Constructive

(23, 58, 120)-net in base 16, using

2 times m-reduction [i] based on (23, 60, 120)-net in base 16, using
- base change [i] based on digital (11, 48, 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]

(58−35, 58, 129)-Net over F₁₆ — Digital

Digital (23, 58, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 58, 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]

(58−35, 58, 5205)-Net in Base 16 — Upper bound on s

There is no (23, 58, 5206)-net in base 16, because

1 times m-reduction [i] would yield (23, 57, 5206)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 432 502962 645974 456880 337892 923656 796764 049384 151891 482849 562653 970131 > 16⁵⁷ [i]