MinT - Best Known (72−40, 72, s)-Nets in Base 128

Best Known (72−40, 72, s)-Nets in Base 128

(72−40, 72, 480)-Net over F₁₂₈ — Constructive and digital

Digital (32, 72, 480)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 23, 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, 49, 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]

(72−40, 72, 545)-Net in Base 128 — Constructive

(32, 72, 545)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (3, 23, 257)-net in base 128, using
  - 1 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, 49, 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]

(72−40, 72, 961)-Net over F₁₂₈ — Digital

Digital (32, 72, 961)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−40, 72, 2520326)-Net in Base 128 — Upper bound on s

There is no (32, 72, 2520327)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 52 374624 842900 987055 656478 875995 959810 200791 718131 398650 995541 443670 351745 526323 859479 951078 283920 247645 707086 652985 110604 281912 801535 366852 060702 823696 > 128⁷² [i]