MinT - Best Known (118−39, 118, s)-Nets in Base 8

Best Known (118−39, 118, s)-Nets in Base 8

(118−39, 118, 388)-Net over F₈ — Constructive and digital

Digital (79, 118, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 26, 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 (53, 92, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 46, 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]

(118−39, 118, 576)-Net in Base 8 — Constructive

(79, 118, 576)-net in base 8, using

4 times m-reduction [i] based on (79, 122, 576)-net in base 8, using
- trace code for nets [i] based on (18, 61, 288)-net in base 64, using
  - 2 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]

(118−39, 118, 1387)-Net over F₈ — Digital

Digital (79, 118, 1387)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(118−39, 118, 412323)-Net in Base 8 — Upper bound on s

There is no (79, 118, 412324)-net in base 8, because

1 times m-reduction [i] would yield (79, 117, 412324)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4587 079188 864753 985363 442231 155954 881321 708148 001709 396878 390665 035261 064933 745298 586037 661339 194549 905778 > 8¹¹⁷ [i]