MinT - Best Known (29, 68, s)-Nets in Base 128

Best Known (29, 68, s)-Nets in Base 128

(29, 68, 438)-Net over F₁₂₈ — Constructive and digital

Digital (29, 68, 438)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. digital (9, 48, 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]

(29, 68, 515)-Net in Base 128 — Constructive

(29, 68, 515)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (3, 22, 257)-net in base 128, using
  - 2 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. (7, 46, 258)-net in base 128, using
  - 2 times m-reduction [i] based on (7, 48, 258)-net in base 128, using
    - base change [i] based on digital (1, 42, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(29, 68, 717)-Net over F₁₂₈ — Digital

Digital (29, 68, 717)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(29, 68, 1683050)-Net in Base 128 — Upper bound on s

There is no (29, 68, 1683051)-net in base 128, because

1 times m-reduction [i] would yield (29, 67, 1683051)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1524 295555 031064 831469 080422 761663 945029 157855 975285 458535 238515 933825 459930 017327 744248 573302 347600 188968 438029 679023 432040 360712 599953 453712 > 128⁶⁷ [i]