MinT - Best Known (68−40, 68, s)-Nets in Base 128

Best Known (68−40, 68, s)-Nets in Base 128

(68−40, 68, 408)-Net over F₁₂₈ — Constructive and digital

Digital (28, 68, 408)-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 (5, 45, 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]

(68−40, 68, 432)-Net in Base 128 — Constructive

(28, 68, 432)-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 (5, 45, 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]

(68−40, 68, 592)-Net over F₁₂₈ — Digital

Digital (28, 68, 592)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−40, 68, 955019)-Net in Base 128 — Upper bound on s

There is no (28, 68, 955020)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 195113 342742 146137 541409 568494 377247 326665 437327 104787 068882 951480 070875 956960 693172 598341 890355 786492 664859 032898 059826 666523 012824 028330 399728 > 128⁶⁸ [i]