MinT - Best Known (32−17, 32, s)-Nets in Base 128

Best Known (32−17, 32, s)-Nets in Base 128

(32−17, 32, 429)-Net over F₁₂₈ — Constructive and digital

Digital (15, 32, 429)-net over F₁₂₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 5, 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, 9, 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, 18, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)

(32−17, 32, 517)-Net in Base 128 — Constructive

(15, 32, 517)-net in base 128, using

base change [i] based on digital (11, 28, 517)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 9, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
  2. digital (2, 19, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(32−17, 32, 885)-Net over F₁₂₈ — Digital

Digital (15, 32, 885)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(32−17, 32, 4338350)-Net in Base 128 — Upper bound on s

There is no (15, 32, 4338351)-net in base 128, because

1 times m-reduction [i] would yield (15, 31, 4338351)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 210624 839731 693756 194189 156927 761459 051152 248552 707137 277770 349791 > 128³¹ [i]