MinT - Best Known (103−77, 103, s)-Nets in Base 16

Best Known (103−77, 103, s)-Nets in Base 16

(103−77, 103, 65)-Net over F₁₆ — Constructive and digital

Digital (26, 103, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 103, 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]

(103−77, 103, 76)-Net in Base 16 — Constructive

(26, 103, 76)-net in base 16, using

2 times m-reduction [i] based on (26, 105, 76)-net in base 16, using
- base change [i] based on digital (5, 84, 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]

(103−77, 103, 150)-Net over F₁₆ — Digital

Digital (26, 103, 150)-net over F₁₆, using

net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
  - a curve from the manYPoints database [i]

(103−77, 103, 1688)-Net in Base 16 — Upper bound on s

There is no (26, 103, 1689)-net in base 16, because

1 times m-reduction [i] would yield (26, 102, 1689)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 670 659368 649212 854572 234733 815146 834502 450637 540876 384699 381732 139723 858942 871881 499679 933795 442234 927739 569604 960596 259856 > 16¹⁰² [i]