MinT - Best Known (31, 31+43, s)-Nets in Base 128

Best Known (31, 31+43, s)-Nets in Base 128

(31, 31+43, 438)-Net over F₁₂₈ — Constructive and digital

Digital (31, 74, 438)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 22, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. digital (9, 52, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(31, 31+43, 514)-Net in Base 128 — Constructive

(31, 74, 514)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (3, 24, 257)-net in base 128, using
  - base change [i] based on digital (0, 21, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]
2. (7, 50, 257)-net in base 128, using
  - 6 times m-reduction [i] based on (7, 56, 257)-net in base 128, using
    - base change [i] based on digital (0, 49, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(31, 31+43, 692)-Net over F₁₂₈ — Digital

Digital (31, 74, 692)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 31+43, 1444604)-Net in Base 128 — Upper bound on s

There is no (31, 74, 1444605)-net in base 128, because

1 times m-reduction [i] would yield (31, 73, 1444605)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 6703 934959 997906 185545 758772 198513 307731 907266 171979 197767 912786 728829 732769 412443 740241 643731 797485 640702 796531 406012 508777 390192 516016 336219 740273 379816 > 128⁷³ [i]