MinT - Best Known (55−32, 55, s)-Nets in Base 128

Best Known (55−32, 55, s)-Nets in Base 128

(55−32, 55, 384)-Net over F₁₂₈ — Constructive and digital

Digital (23, 55, 384)-net over F₁₂₈, using

2 times m-reduction [i] based on digital (23, 57, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 20, 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, 37, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(55−32, 55, 514)-Net in Base 128 — Constructive

(23, 55, 514)-net in base 128, using

1 times m-reduction [i] based on (23, 56, 514)-net in base 128, using
- base change [i] based on digital (16, 49, 514)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 16, 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. digital (0, 33, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(55−32, 55, 551)-Net over F₁₂₈ — Digital

Digital (23, 55, 551)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(55−32, 55, 938135)-Net in Base 128 — Upper bound on s

There is no (23, 55, 938136)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 78 804180 396429 832080 647930 977895 905247 707314 594071 105344 968802 620596 243517 043276 853831 911070 144302 750597 647112 970653 > 128⁵⁵ [i]