MinT - Best Known (33−18, 33, s)-Nets in Base 128

Best Known (33−18, 33, s)-Nets in Base 128

(33−18, 33, 387)-Net over F₁₂₈ — Constructive and digital

Digital (15, 33, 387)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (15, 34, 387)-net over F₁₂₈, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 6, 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, 9, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
  3. digital (0, 19, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)

(33−18, 33, 515)-Net in Base 128 — Constructive

(15, 33, 515)-net in base 128, using

1 times m-reduction [i] based on (15, 34, 515)-net in base 128, using
- (u,Â u+v)-construction [i] based on
  1. (2, 11, 257)-net in base 128, using
    - 5 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. (4, 23, 258)-net in base 128, using
    - 1 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]

(33−18, 33, 705)-Net over F₁₂₈ — Digital

Digital (15, 33, 705)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(33−18, 33, 1739335)-Net in Base 128 — Upper bound on s

There is no (15, 33, 1739336)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 3450 888893 474828 540778 498074 374117 331542 678262 148045 524417 929994 708872 > 128³³ [i]