MinT - Best Known (48−25, 48, s)-Nets in Base 128

Best Known (48−25, 48, s)-Nets in Base 128

(48−25, 48, 450)-Net over F₁₂₈ — Constructive and digital

Digital (23, 48, 450)-net over F₁₂₈, using

generalized (u,Â u+v)-construction [i] based on
1. 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]
2. digital (1, 13, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)
3. digital (1, 26, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)

(48−25, 48, 545)-Net in Base 128 — Constructive

(23, 48, 545)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (2, 14, 257)-net in base 128, using
  - 2 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. digital (9, 34, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(48−25, 48, 1277)-Net over F₁₂₈ — Digital

Digital (23, 48, 1277)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(48−25, 48, 7460989)-Net in Base 128 — Upper bound on s

There is no (23, 48, 7460990)-net in base 128, because

1 times m-reduction [i] would yield (23, 47, 7460990)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1093 626663 145011 736027 763406 627562 011067 761978 373182 235596 205170 708800 217899 729270 634286 559768 462827 > 128⁴⁷ [i]