MinT - Best Known (119−41, 119, s)-Nets in Base 8

Best Known (119−41, 119, s)-Nets in Base 8

(119−41, 119, 378)-Net over F₈ — Constructive and digital

Digital (78, 119, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 23, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (55, 96, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 48, 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]

(119−41, 119, 576)-Net in Base 8 — Constructive

(78, 119, 576)-net in base 8, using

1 times m-reduction [i] based on (78, 120, 576)-net in base 8, using
- trace code for nets [i] based on (18, 60, 288)-net in base 64, using
  - 3 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 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]

(119−41, 119, 1115)-Net over F₈ — Digital

Digital (78, 119, 1115)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(119−41, 119, 252590)-Net in Base 8 — Upper bound on s

There is no (78, 119, 252591)-net in base 8, because

1 times m-reduction [i] would yield (78, 118, 252591)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 36695 984110 237842 801969 442074 757592 464133 321850 071129 084838 330985 591641 241630 468664 608431 647092 845766 432018 > 8¹¹⁸ [i]