MinT - Best Known (30, 30+93, s)-Nets in Base 16

Best Known (30, 30+93, s)-Nets in Base 16

(30, 30+93, 65)-Net over F₁₆ — Constructive and digital

Digital (30, 123, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 123, 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, 30+93, 76)-Net in Base 16 — Constructive

(30, 123, 76)-net in base 16, using

2 times m-reduction [i] based on (30, 125, 76)-net in base 16, using
- base change [i] based on digital (5, 100, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(30, 30+93, 162)-Net over F₁₆ — Digital

Digital (30, 123, 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, 30+93, 1848)-Net in Base 16 — Upper bound on s

There is no (30, 123, 1849)-net in base 16, because

1 times m-reduction [i] would yield (30, 122, 1849)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 816 870754 594605 266964 847593 614617 999775 243454 518446 656091 708770 556780 926024 748433 536746 960186 410020 135560 745997 326266 848853 928296 004600 976008 678936 > 16¹²² [i]