MinT - Best Known (103−75, 103, s)-Nets in Base 16

Best Known (103−75, 103, s)-Nets in Base 16

(103−75, 103, 65)-Net over F₁₆ — Constructive and digital

Digital (28, 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]

(103−75, 103, 98)-Net in Base 16 — Constructive

(28, 103, 98)-net in base 16, using

2 times m-reduction [i] based on (28, 105, 98)-net in base 16, using
- base change [i] based on digital (7, 84, 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]

(103−75, 103, 156)-Net over F₁₆ — Digital

Digital (28, 103, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 103, 156)-net over F₁₆, using
- net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
    - a curve from the manYPoints database [i]

(103−75, 103, 2017)-Net in Base 16 — Upper bound on s

There is no (28, 103, 2018)-net in base 16, because

1 times m-reduction [i] would yield (28, 102, 2018)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 661 410423 710231 000156 874768 702492 696421 666967 823767 682917 881028 733555 793785 706028 945459 756678 439269 450067 810520 781203 891216 > 16¹⁰² [i]