MinT - Best Known (156−51, 156, s)-Nets in Base 8

Best Known (156−51, 156, s)-Nets in Base 8

(156−51, 156, 419)-Net over F₈ — Constructive and digital

Digital (105, 156, 419)-net over F₈, using

8¹ times duplication [i] based on digital (104, 155, 419)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 39, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
  2. digital (65, 116, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 58, 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]

(156−51, 156, 576)-Net in Base 8 — Constructive

(105, 156, 576)-net in base 8, using

12 times m-reduction [i] based on (105, 168, 576)-net in base 8, using
- trace code for nets [i] based on (21, 84, 288)-net in base 64, using
  - base change [i] based on digital (9, 72, 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]

(156−51, 156, 1854)-Net over F₈ — Digital

Digital (105, 156, 1854)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(156−51, 156, 577662)-Net in Base 8 — Upper bound on s

There is no (105, 156, 577663)-net in base 8, because

1 times m-reduction [i] would yield (105, 155, 577663)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 95 271229 304289 721017 188886 019160 252556 276645 742010 361707 263389 777344 644280 018481 774852 572191 714379 615809 465901 297999 179162 966666 731618 373250 > 8¹⁵⁵ [i]