MinT - Best Known (42−22, 42, s)-Nets in Base 128

Best Known (42−22, 42, s)-Nets in Base 128

(42−22, 42, 429)-Net over F₁₂₈ — Constructive and digital

Digital (20, 42, 429)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (20, 43, 429)-net over F₁₂₈, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 7, 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 (1, 12, 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]
  3. digital (1, 24, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)

(42−22, 42, 517)-Net in Base 128 — Constructive

(20, 42, 517)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (3, 14, 258)-net in base 128, using
  - 2 times m-reduction [i] based on (3, 16, 258)-net in base 128, using
    - base change [i] based on digital (1, 14, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. (6, 28, 259)-net in base 128, using
  - 4 times m-reduction [i] based on (6, 32, 259)-net in base 128, using
    - base change [i] based on digital (2, 28, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(42−22, 42, 1130)-Net over F₁₂₈ — Digital

Digital (20, 42, 1130)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(42−22, 42, 4294580)-Net in Base 128 — Upper bound on s

There is no (20, 42, 4294581)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 31828 755116 903808 406351 756016 872590 262927 170761 783042 215936 266953 611472 572765 816330 017528 > 128⁴² [i]