MinT - Best Known (34, 34+41, s)-Nets in Base 128

Best Known (34, 34+41, s)-Nets in Base 128

(34, 34+41, 504)-Net over F₁₂₈ — Constructive and digital

Digital (34, 75, 504)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 25, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]
2. digital (9, 50, 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]

(34, 34+41, 546)-Net in Base 128 — Constructive

(34, 75, 546)-net in base 128, using

128¹ times duplication [i] based on (33, 74, 546)-net in base 128, using
- (u,Â u+v)-construction [i] based on
  1. (4, 24, 258)-net in base 128, using
    - base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
  2. digital (9, 50, 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]

(34, 34+41, 1129)-Net over F₁₂₈ — Digital

Digital (34, 75, 1129)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(34, 34+41, 4094287)-Net in Base 128 — Upper bound on s

There is no (34, 75, 4094288)-net in base 128, because

1 times m-reduction [i] would yield (34, 74, 4094288)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 858100 312679 987898 159104 031351 278103 922702 955837 257338 271742 379094 943580 951983 120639 002971 300851 480689 994814 161328 984792 024294 341676 641709 671116 162988 576324 > 128⁷⁴ [i]