MinT - Best Known (65−34, 65, s)-Nets in Base 128

Best Known (65−34, 65, s)-Nets in Base 128

(65−34, 65, 504)-Net over F₁₂₈ — Constructive and digital

Digital (31, 65, 504)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 22, 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, 43, 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]

(65−34, 65, 547)-Net in Base 128 — Constructive

(31, 65, 547)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (5, 22, 259)-net in base 128, using
  - 2 times m-reduction [i] based on (5, 24, 259)-net in base 128, using
    - base change [i] based on digital (2, 21, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. digital (9, 43, 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]

(65−34, 65, 1482)-Net over F₁₂₈ — Digital

Digital (31, 65, 1482)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(65−34, 65, 6443476)-Net in Base 128 — Upper bound on s

There is no (31, 65, 6443477)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 93035 536390 007185 071933 004476 460395 315744 762434 610185 532653 047430 113896 592612 270554 349098 172629 936610 550592 729838 240526 255614 380125 391292 > 128⁶⁵ [i]