MinT - Best Known (77−47, 77, s)-Nets in Base 16

Best Known (77−47, 77, s)-Nets in Base 16

(77−47, 77, 89)-Net over F₁₆ — Constructive and digital

Digital (30, 77, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 24, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (6, 53, 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]

(77−47, 77, 120)-Net in Base 16 — Constructive

(30, 77, 120)-net in base 16, using

18 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]

(77−47, 77, 162)-Net over F₁₆ — Digital

Digital (30, 77, 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]

(77−47, 77, 5974)-Net in Base 16 — Upper bound on s

There is no (30, 77, 5975)-net in base 16, because

1 times m-reduction [i] would yield (30, 76, 5975)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 32 681058 218167 944769 314679 653070 789679 726437 219529 288476 449424 754788 155351 774915 609450 651376 > 16⁷⁶ [i]