MinT - Best Known (73−43, 73, s)-Nets in Base 16

Best Known (73−43, 73, s)-Nets in Base 16

(73−43, 73, 103)-Net over F₁₆ — Constructive and digital

Digital (30, 73, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 24, 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]
2. digital (6, 49, 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]

(73−43, 73, 128)-Net in Base 16 — Constructive

(30, 73, 128)-net in base 16, using

2 times m-reduction [i] based on (30, 75, 128)-net in base 16, using
- base change [i] based on digital (5, 50, 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]

(73−43, 73, 162)-Net over F₁₆ — Digital

Digital (30, 73, 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]

(73−43, 73, 7765)-Net in Base 16 — Upper bound on s

There is no (30, 73, 7766)-net in base 16, because

1 times m-reduction [i] would yield (30, 72, 7766)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 497 841006 184887 023487 557396 076603 498873 659388 561082 097180 189594 639555 763732 428179 192291 > 16⁷² [i]