MinT - Best Known (34, 77, s)-Nets in Base 128

Best Known (34, 77, s)-Nets in Base 128

(34, 77, 480)-Net over F₁₂₈ — Constructive and digital

Digital (34, 77, 480)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (34, 78, 480)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 25, 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, 53, 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]

(34, 77, 545)-Net in Base 128 — Constructive

(34, 77, 545)-net in base 128, using

128¹ times duplication [i] based on (33, 76, 545)-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. 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]

(34, 77, 970)-Net over F₁₂₈ — Digital

Digital (34, 77, 970)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(34, 77, 2889219)-Net in Base 128 — Upper bound on s

There is no (34, 77, 2889220)-net in base 128, because

1 times m-reduction [i] would yield (34, 76, 2889220)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 14059 153714 252336 960090 488402 264599 376338 325380 030482 215974 810392 777785 899616 755982 425160 254208 160878 599026 685055 332808 609426 144356 272411 622759 535195 582043 997568 > 128⁷⁶ [i]