MinT - Best Known (144−47, 144, s)-Nets in Base 8

Best Known (144−47, 144, s)-Nets in Base 8

(144−47, 144, 402)-Net over F₈ — Constructive and digital

Digital (97, 144, 402)-net over F₈, using

1 times m-reduction [i] based on digital (97, 145, 402)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 35, 48)-net over F₈, using
    - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
        a function field by SÃ©mirat [i]
  2. digital (62, 110, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 55, 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]

(144−47, 144, 576)-Net in Base 8 — Constructive

(97, 144, 576)-net in base 8, using

10 times m-reduction [i] based on (97, 154, 576)-net in base 8, using
- trace code for nets [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]

(144−47, 144, 1750)-Net over F₈ — Digital

Digital (97, 144, 1750)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(144−47, 144, 554895)-Net in Base 8 — Upper bound on s

There is no (97, 144, 554896)-net in base 8, because

1 times m-reduction [i] would yield (97, 143, 554896)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1386 386473 018796 527917 618106 798610 295269 024389 762642 071280 149195 825849 060532 812050 235233 939116 325612 481024 579292 097856 284860 565912 > 8¹⁴³ [i]