MinT - Best Known (63, 63+81, s)-Nets in Base 8

Best Known (63, 63+81, s)-Nets in Base 8

(63, 63+81, 110)-Net over F₈ — Constructive and digital

Digital (63, 144, 110)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 49, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]
2. digital (14, 95, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(63, 63+81, 147)-Net over F₈ — Digital

Digital (63, 144, 147)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(63, 63+81, 156)-Net in Base 8

(63, 144, 156)-net in base 8, using

base change [i] based on digital (27, 108, 156)-net over F₁₆, using
- net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
    - a curve from the manYPoints database [i]

(63, 63+81, 3787)-Net in Base 8 — Upper bound on s

There is no (63, 144, 3788)-net in base 8, because

1 times m-reduction [i] would yield (63, 143, 3788)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1392 175187 550490 537434 913044 290430 314043 306737 696017 740156 086867 165037 104028 836697 065433 690976 068738 021705 157416 488228 933642 349810 > 8¹⁴³ [i]