MinT - Best Known (30, 78, s)-Nets in Base 16

Best Known (30, 78, s)-Nets in Base 16

(30, 78, 82)-Net over F₁₆ — Constructive and digital

Digital (30, 78, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 24, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (6, 54, 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]

(30, 78, 120)-Net in Base 16 — Constructive

(30, 78, 120)-net in base 16, using

17 times m-reduction [i] based on (30, 95, 120)-net in base 16, using
- base change [i] based on digital (11, 76, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(30, 78, 162)-Net over F₁₆ — Digital

Digital (30, 78, 162)-net over F₁₆, using

net from sequence [i] based on digital (30, 161)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 30 and N(F) ≥ 162, using
  - a curve from the manYPoints database [i]

(30, 78, 5340)-Net in Base 16 — Upper bound on s

There is no (30, 78, 5341)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8354 956469 018982 071725 276041 721758 780760 854846 365432 052779 716756 896304 361597 675411 559785 548111 > 16⁷⁸ [i]