MinT - Best Known (140−49, 140, s)-Nets in Base 8

Best Known (140−49, 140, s)-Nets in Base 8

(140−49, 140, 379)-Net over F₈ — Constructive and digital

Digital (91, 140, 379)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 28, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (63, 112, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 56, 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]

(140−49, 140, 576)-Net in Base 8 — Constructive

(91, 140, 576)-net in base 8, using

t-expansion [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]

(140−49, 140, 1177)-Net over F₈ — Digital

Digital (91, 140, 1177)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(140−49, 140, 238031)-Net in Base 8 — Upper bound on s

There is no (91, 140, 238032)-net in base 8, because

1 times m-reduction [i] would yield (91, 139, 238032)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 338492 657836 951637 858684 119807 907239 530221 162482 540616 242464 040144 672936 286078 441963 670751 308820 285104 589619 234630 594460 787684 > 8¹³⁹ [i]