MinT - Best Known (47−27, 47, s)-Nets in Base 128

Best Known (47−27, 47, s)-Nets in Base 128

(47−27, 47, 384)-Net over F₁₂₈ — Constructive and digital

Digital (20, 47, 384)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (20, 48, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 17, 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 (3, 31, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(47−27, 47, 515)-Net in Base 128 — Constructive

(20, 47, 515)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (2, 15, 257)-net in base 128, using
  - 1 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
    - base change [i] based on digital (0, 14, 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. (5, 32, 258)-net in base 128, using
  - base change [i] based on digital (1, 28, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(47−27, 47, 548)-Net over F₁₂₈ — Digital

Digital (20, 47, 548)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(47−27, 47, 1276091)-Net in Base 128 — Upper bound on s

There is no (20, 47, 1276092)-net in base 128, because

1 times m-reduction [i] would yield (20, 46, 1276092)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 8 543992 786460 454441 347393 597797 665502 952921 539044 887641 322362 808788 739403 065935 565070 446995 358172 > 128⁴⁶ [i]