MinT - Best Known (28, 61, s)-Nets in Base 128

Best Known (28, 61, s)-Nets in Base 128

(28, 61, 480)-Net over F₁₂₈ — Constructive and digital

Digital (28, 61, 480)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 19, 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, 42, 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]

(28, 61, 545)-Net in Base 128 — Constructive

(28, 61, 545)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (3, 19, 257)-net in base 128, using
  - 5 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, 42, 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]

(28, 61, 1063)-Net over F₁₂₈ — Digital

Digital (28, 61, 1063)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(28, 61, 4273381)-Net in Base 128 — Upper bound on s

There is no (28, 61, 4273382)-net in base 128, because

1 times m-reduction [i] would yield (28, 60, 4273382)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 2 707695 378761 206816 352211 604218 433027 192678 958103 209562 227370 506656 957271 726395 932547 821913 122700 151944 489840 183601 555431 543410 > 128⁶⁰ [i]