MinT - Best Known (136−46, 136, s)-Nets in Base 8

Best Known (136−46, 136, s)-Nets in Base 8

(136−46, 136, 388)-Net over F₈ — Constructive and digital

Digital (90, 136, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 30, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (60, 106, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 53, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(136−46, 136, 576)-Net in Base 8 — Constructive

(90, 136, 576)-net in base 8, using

t-expansion [i] based on (89, 136, 576)-net in base 8, using
- 4 times m-reduction [i] based on (89, 140, 576)-net in base 8, using
  - trace code for nets [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 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]

(136−46, 136, 1373)-Net over F₈ — Digital

Digital (90, 136, 1373)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(136−46, 136, 294677)-Net in Base 8 — Upper bound on s

There is no (90, 136, 294678)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 661 061907 111159 381772 012062 143318 676536 602317 417816 676945 455805 787098 532647 122521 092952 456662 405480 811828 101115 378295 317056 > 8¹³⁶ [i]