MinT - Best Known (139−49, 139, s)-Nets in Base 8

Best Known (139−49, 139, s)-Nets in Base 8

(139−49, 139, 378)-Net over F₈ — Constructive and digital

Digital (90, 139, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 27, 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 (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]

(139−49, 139, 576)-Net in Base 8 — Constructive

(90, 139, 576)-net in base 8, using

t-expansion [i] based on (89, 139, 576)-net in base 8, using
- 1 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]

(139−49, 139, 1128)-Net over F₈ — Digital

Digital (90, 139, 1128)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−49, 139, 218274)-Net in Base 8 — Upper bound on s

There is no (90, 139, 218275)-net in base 8, because

1 times m-reduction [i] would yield (90, 138, 218275)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 42311 268449 341715 000641 458689 214950 175666 844196 063588 776182 750276 847322 990707 017935 331726 412249 312529 136774 343521 839814 357916 > 8¹³⁸ [i]