MinT - Best Known (59−34, 59, s)-Nets in Base 128

Best Known (59−34, 59, s)-Nets in Base 128

(59−34, 59, 408)-Net over F₁₂₈ — Constructive and digital

Digital (25, 59, 408)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 20, 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 (5, 39, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(59−34, 59, 514)-Net in Base 128 — Constructive

(25, 59, 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, 39, 257)-net in base 128, using
  - 1 times m-reduction [i] based on (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)

(59−34, 59, 623)-Net over F₁₂₈ — Digital

Digital (25, 59, 623)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(59−34, 59, 1162509)-Net in Base 128 — Upper bound on s

There is no (25, 59, 1162510)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 21153 890615 034636 315452 296985 951454 280918 008829 756755 074004 500365 318177 844401 199700 786492 774165 864719 333907 673092 052419 506416 > 128⁵⁹ [i]