MinT - Best Known (31, 31+42, s)-Nets in Base 16

Best Known (31, 31+42, s)-Nets in Base 16

(31, 31+42, 110)-Net over F₁₆ — Constructive and digital

Digital (31, 73, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 25, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (6, 48, 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+42, 128)-Net in Base 16 — Constructive

(31, 73, 128)-net in base 16, using

5 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+42, 168)-Net over F₁₆ — Digital

Digital (31, 73, 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+42, 8863)-Net in Base 16 — Upper bound on s

There is no (31, 73, 8864)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7971 046010 028252 360672 019530 334438 918018 545318 802825 633570 666668 481846 387883 870534 953411 > 16⁷³ [i]