MinT - Best Known (12, 26, s)-Nets in Base 128

Best Known (12, 26, s)-Nets in Base 128

(12, 26, 408)-Net over F₁₂₈ — Constructive and digital

Digital (12, 26, 408)-net over F₁₂₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 4, 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 (0, 7, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
3. digital (1, 15, 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]

(12, 26, 515)-Net in Base 128 — Constructive

(12, 26, 515)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (1, 8, 257)-net in base 128, using
  - base change [i] based on digital (0, 7, 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. (4, 18, 258)-net in base 128, using
  - 6 times m-reduction [i] based on (4, 24, 258)-net in base 128, using
    - base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(12, 26, 738)-Net over F₁₂₈ — Digital

Digital (12, 26, 738)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(12, 26, 1786051)-Net in Base 128 — Upper bound on s

There is no (12, 26, 1786052)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 6 129984 881327 678822 965308 881379 674842 537622 053372 508688 > 128²⁶ [i]