MinT - Best Known (58−41, 58, s)-Nets in Base 16

Best Known (58−41, 58, s)-Nets in Base 16

(58−41, 58, 65)-Net over F₁₆ — Constructive and digital

Digital (17, 58, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 58, 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]

(58−41, 58, 76)-Net in Base 16 — Constructive

(17, 58, 76)-net in base 16, using

2 times m-reduction [i] based on (17, 60, 76)-net in base 16, using
- base change [i] based on digital (5, 48, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(58−41, 58, 112)-Net over F₁₆ — Digital

Digital (17, 58, 112)-net over F₁₆, using

net from sequence [i] based on digital (17, 111)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 17 and N(F) ≥ 112, using
  - a curve from the manYPoints database [i]

(58−41, 58, 1485)-Net in Base 16 — Upper bound on s

There is no (17, 58, 1486)-net in base 16, because

1 times m-reduction [i] would yield (17, 57, 1486)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 435 564913 587403 919980 201171 091089 045126 479648 382782 262578 206317 986676 > 16⁵⁷ [i]