MinT - Best Known (89, 136, s)-Nets in Base 8

Best Known (89, 136, s)-Nets in Base 8

(89, 136, 382)-Net over F₈ — Constructive and digital

Digital (89, 136, 382)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 28, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (61, 108, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 54, 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]

(89, 136, 576)-Net in Base 8 — Constructive

(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]

(89, 136, 1226)-Net over F₈ — Digital

Digital (89, 136, 1226)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 136, 269203)-Net in Base 8 — Upper bound on s

There is no (89, 136, 269204)-net in base 8, because

1 times m-reduction [i] would yield (89, 135, 269204)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 82 635229 295302 816520 754283 521750 429334 207525 013579 244829 732896 396664 972846 533294 636490 256826 356061 840819 185124 629160 686880 > 8¹³⁵ [i]