MinT - Best Known (31, 31+45, s)-Nets in Base 16

Best Known (31, 31+45, s)-Nets in Base 16

(31, 31+45, 103)-Net over F₁₆ — Constructive and digital

Digital (31, 76, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 25, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (6, 51, 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]

(31, 31+45, 128)-Net in Base 16 — Constructive

(31, 76, 128)-net in base 16, using

2 times m-reduction [i] based on (31, 78, 128)-net in base 16, using
- base change [i] based on digital (5, 52, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(31, 31+45, 168)-Net over F₁₆ — Digital

Digital (31, 76, 168)-net over F₁₆, using

net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
  - a curve from the manYPoints database [i]

(31, 31+45, 7674)-Net in Base 16 — Upper bound on s

There is no (31, 76, 7675)-net in base 16, because

1 times m-reduction [i] would yield (31, 75, 7675)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 041010 087925 369179 788884 613660 218894 872766 653802 530302 058571 406485 623756 091524 696498 369626 > 16⁷⁵ [i]