MinT - Best Known (30, 30+39, s)-Nets in Base 16

Best Known (30, 30+39, s)-Nets in Base 16

(30, 30+39, 114)-Net over F₁₆ — Constructive and digital

Digital (30, 69, 114)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 24, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]
2. digital (6, 45, 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]

(30, 30+39, 162)-Net over F₁₆ — Digital

Digital (30, 69, 162)-net over F₁₆, using

net from sequence [i] based on digital (30, 161)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 30 and N(F) ≥ 162, using
  - a curve from the manYPoints database [i]

(30, 30+39, 177)-Net in Base 16 — Constructive

(30, 69, 177)-net in base 16, using

base change [i] based on digital (7, 46, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(30, 30+39, 10769)-Net in Base 16 — Upper bound on s

There is no (30, 69, 10770)-net in base 16, because

1 times m-reduction [i] would yield (30, 68, 10770)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7596 227231 671731 309235 824749 516659 873703 776350 695465 281698 503496 992444 024554 648576 > 16⁶⁸ [i]