MinT - Best Known (139−43, 139, s)-Nets in Base 8

Best Known (139−43, 139, s)-Nets in Base 8

(139−43, 139, 419)-Net over F₈ — Constructive and digital

Digital (96, 139, 419)-net over F₈, using

t-expansion [i] based on digital (95, 139, 419)-net over F₈, using
- 1 times m-reduction [i] based on digital (95, 140, 419)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (14, 36, 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 (59, 104, 354)-net over F₈, using
      - trace code for nets [i] based on digital (7, 52, 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]

(139−43, 139, 576)-Net in Base 8 — Constructive

(96, 139, 576)-net in base 8, using

13 times m-reduction [i] based on (96, 152, 576)-net in base 8, using
- trace code for nets [i] based on (20, 76, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (20, 77, 288)-net in base 64, using
    - base change [i] based on digital (9, 66, 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]

(139−43, 139, 2320)-Net over F₈ — Digital

Digital (96, 139, 2320)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−43, 139, 1066548)-Net in Base 8 — Upper bound on s

There is no (96, 139, 1066549)-net in base 8, because

1 times m-reduction [i] would yield (96, 138, 1066549)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 42308 064901 054490 647298 707114 254669 435882 250247 579529 720566 956040 461860 827228 117344 894748 204361 868011 605331 078442 441280 080824 > 8¹³⁸ [i]