MinT - Best Known (63−39, 63, s)-Nets in Base 128

Best Known (63−39, 63, s)-Nets in Base 128

(63−39, 63, 345)-Net over F₁₂₈ — Constructive and digital

Digital (24, 63, 345)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 19, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. digital (5, 44, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(63−39, 63, 513)-Net over F₁₂₈ — Digital

Digital (24, 63, 513)-net over F₁₂₈, using

net from sequence [i] based on digital (24, 512)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 24 and N(F) ≥ 513, using
  - a curve from the manYPoints database [i]

(63−39, 63, 469420)-Net in Base 128 — Upper bound on s

There is no (24, 63, 469421)-net in base 128, because

1 times m-reduction [i] would yield (24, 62, 469421)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 44363 385399 316826 481772 926186 784262 268096 578172 682274 390551 751719 742932 273255 322326 029847 828994 434569 020316 108929 572860 168812 835816 > 128⁶² [i]