MinT - Best Known (49−28, 49, s)-Nets in Base 128

Best Known (49−28, 49, s)-Nets in Base 128

(49−28, 49, 384)-Net over F₁₂₈ — Constructive and digital

Digital (21, 49, 384)-net over F₁₂₈, using

2 times m-reduction [i] based on digital (21, 51, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 18, 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, 33, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(49−28, 49, 514)-Net in Base 128 — Constructive

(21, 49, 514)-net in base 128, using

1 times m-reduction [i] based on (21, 50, 514)-net in base 128, using
- (u,Â u+v)-construction [i] based on
  1. (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, 34, 257)-net in base 128, using
    - 6 times m-reduction [i] based on (5, 40, 257)-net in base 128, using
      - base change [i] based on digital (0, 35, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(49−28, 49, 587)-Net over F₁₂₈ — Digital

Digital (21, 49, 587)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(49−28, 49, 1129500)-Net in Base 128 — Upper bound on s

There is no (21, 49, 1129501)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 17 918101 846414 177458 786129 303990 607225 935316 814356 393923 151331 845396 381880 908208 662307 778795 022125 844344 > 128⁴⁹ [i]