MinT - Best Known (87−55, 87, s)-Nets in Base 16

Best Known (87−55, 87, s)-Nets in Base 16

(87−55, 87, 71)-Net over F₁₆ — Constructive and digital

Digital (32, 87, 71)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 29, 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 (3, 58, 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]

(87−55, 87, 120)-Net in Base 16 — Constructive

(32, 87, 120)-net in base 16, using

18 times m-reduction [i] based on (32, 105, 120)-net in base 16, using
- base change [i] based on digital (11, 84, 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]

(87−55, 87, 168)-Net over F₁₆ — Digital

Digital (32, 87, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 87, 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]

(87−55, 87, 4970)-Net in Base 16 — Upper bound on s

There is no (32, 87, 4971)-net in base 16, because

1 times m-reduction [i] would yield (32, 86, 4971)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 35 999476 150765 884578 577749 175113 826609 358793 895363 935935 547262 404478 211016 204722 230033 025499 279204 745856 > 16⁸⁶ [i]