MinT - Best Known (81−51, 81, s)-Nets in Base 16

Best Known (81−51, 81, s)-Nets in Base 16

(81−51, 81, 71)-Net over F₁₆ — Constructive and digital

Digital (30, 81, 71)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 27, 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, 54, 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]

(81−51, 81, 120)-Net in Base 16 — Constructive

(30, 81, 120)-net in base 16, using

14 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]

(81−51, 81, 162)-Net over F₁₆ — Digital

Digital (30, 81, 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]

(81−51, 81, 4824)-Net in Base 16 — Upper bound on s

There is no (30, 81, 4825)-net in base 16, because

1 times m-reduction [i] would yield (30, 80, 4825)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 136763 641005 181437 811310 316671 176818 802644 956308 617125 163640 710983 799588 711820 511742 708978 559376 > 16⁸⁰ [i]