MinT - Best Known (26, 61, s)-Nets in Base 128

Best Known (26, 61, s)-Nets in Base 128

(26, 61, 417)-Net over F₁₂₈ — Constructive and digital

Digital (26, 61, 417)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 17, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. digital (9, 44, 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]

(26, 61, 514)-Net in Base 128 — Constructive

(26, 61, 514)-net in base 128, using

128¹ times duplication [i] based on (25, 60, 514)-net in base 128, using
- (u,Â u+v)-construction [i] based on
  1. (3, 20, 257)-net in base 128, using
    - 4 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. (5, 40, 257)-net in base 128, using
    - base change [i] based on digital (0, 35, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(26, 61, 660)-Net over F₁₂₈ — Digital

Digital (26, 61, 660)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(26, 61, 1546503)-Net in Base 128 — Upper bound on s

There is no (26, 61, 1546504)-net in base 128, because

1 times m-reduction [i] would yield (26, 60, 1546504)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 2 707705 217149 555484 030734 097889 979227 516785 478315 496519 135102 753922 813712 447610 250452 861548 494078 525064 890400 832089 893867 026864 > 128⁶⁰ [i]