MinT - Best Known (28, 67, s)-Nets in Base 128

Best Known (28, 67, s)-Nets in Base 128

(28, 67, 417)-Net over F₁₂₈ — Constructive and digital

Digital (28, 67, 417)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 19, 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, 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]

(28, 67, 514)-Net in Base 128 — Constructive

(28, 67, 514)-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. (6, 45, 257)-net in base 128, using
  - 3 times m-reduction [i] based on (6, 48, 257)-net in base 128, using
    - base change [i] based on digital (0, 42, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(28, 67, 633)-Net over F₁₂₈ — Digital

Digital (28, 67, 633)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(28, 67, 1303739)-Net in Base 128 — Upper bound on s

There is no (28, 67, 1303740)-net in base 128, because

1 times m-reduction [i] would yield (28, 66, 1303740)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 11 908657 417845 968504 315526 879730 126460 539693 257655 669145 005156 460069 933459 311432 499969 012105 535106 377688 241477 120367 366237 239724 266354 818331 > 128⁶⁶ [i]