MinT - Best Known (32, 32+39, s)-Nets in Base 128

Best Known (32, 32+39, s)-Nets in Base 128

(32, 32+39, 480)-Net over F₁₂₈ — Constructive and digital

Digital (32, 71, 480)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (32, 72, 480)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 23, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]
  2. digital (9, 49, 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]

(32, 32+39, 546)-Net in Base 128 — Constructive

(32, 71, 546)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (4, 23, 258)-net in base 128, using
  - 1 times m-reduction [i] based on (4, 24, 258)-net in base 128, using
    - base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. digital (9, 48, 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]

(32, 32+39, 1043)-Net over F₁₂₈ — Digital

Digital (32, 71, 1043)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(32, 32+39, 3620894)-Net in Base 128 — Upper bound on s

There is no (32, 71, 3620895)-net in base 128, because

1 times m-reduction [i] would yield (32, 70, 3620895)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 3196 683357 935302 067996 157765 388234 613598 442678 991471 220901 002300 623055 944580 116313 999667 072083 504309 948212 695432 490989 154152 002192 181700 023060 889500 > 128⁷⁰ [i]