MinT - Best Known (31, 77, s)-Nets in Base 16

Best Known (31, 77, s)-Nets in Base 16

(31, 77, 98)-Net over F₁₆ — Constructive and digital

Digital (31, 77, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 25, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (6, 52, 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, 77, 128)-Net in Base 16 — Constructive

(31, 77, 128)-net in base 16, using

1 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, 77, 168)-Net over F₁₆ — Digital

Digital (31, 77, 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, 77, 6741)-Net in Base 16 — Upper bound on s

There is no (31, 77, 6742)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 522 743704 912906 178380 235361 005202 934076 129327 565149 303188 240979 882243 910216 471657 925456 554016 > 16⁷⁷ [i]