MinT - Best Known (27, 27+31, s)-Nets in Base 128

Best Known (27, 27+31, s)-Nets in Base 128

(27, 27+31, 480)-Net over F₁₂₈ — Constructive and digital

Digital (27, 58, 480)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 18, 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, 40, 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]

(27, 27+31, 545)-Net in Base 128 — Constructive

(27, 58, 545)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (3, 18, 257)-net in base 128, using
  - 6 times m-reduction [i] based on (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. digital (9, 40, 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]

(27, 27+31, 1139)-Net over F₁₂₈ — Digital

Digital (27, 58, 1139)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(27, 27+31, 5144701)-Net in Base 128 — Upper bound on s

There is no (27, 58, 5144702)-net in base 128, because

1 times m-reduction [i] would yield (27, 57, 5144702)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1 291125 845720 266104 838487 253088 092419 379125 523245 139489 706228 215618 178476 669871 967904 533962 810623 665847 165015 056936 396232 > 128⁵⁷ [i]