MinT - Best Known (25, 25+27, s)-Nets in Base 128

Best Known (25, 25+27, s)-Nets in Base 128

(25, 25+27, 480)-Net over F₁₂₈ — Constructive and digital

Digital (25, 52, 480)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 16, 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 (9, 36, 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]

(25, 25+27, 546)-Net in Base 128 — Constructive

(25, 52, 546)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (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. digital (9, 36, 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]

(25, 25+27, 1374)-Net over F₁₂₈ — Digital

Digital (25, 52, 1374)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(25, 25+27, 8248005)-Net in Base 128 — Upper bound on s

There is no (25, 52, 8248006)-net in base 128, because

1 times m-reduction [i] would yield (25, 51, 8248006)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 293568 071428 805481 493525 820804 112160 265378 640656 964558 914456 737165 699708 985056 600970 200608 407523 791423 431912 > 128⁵¹ [i]