MinT - Best Known (140−47, 140, s)-Nets in Base 8

Best Known (140−47, 140, s)-Nets in Base 8

(140−47, 140, 399)-Net over F₈ — Constructive and digital

Digital (93, 140, 399)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 32, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]
2. digital (61, 108, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 54, 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]

(140−47, 140, 576)-Net in Base 8 — Constructive

(93, 140, 576)-net in base 8, using

6 times m-reduction [i] based on (93, 146, 576)-net in base 8, using
- trace code for nets [i] based on (20, 73, 288)-net in base 64, using
  - 4 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]

(140−47, 140, 1464)-Net over F₈ — Digital

Digital (93, 140, 1464)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(140−47, 140, 386497)-Net in Base 8 — Upper bound on s

There is no (93, 140, 386498)-net in base 8, because

1 times m-reduction [i] would yield (93, 139, 386498)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 338467 097047 048607 913538 658416 071554 884426 903818 579847 475145 684646 118802 440576 583277 457614 681066 270862 273993 386959 887331 786976 > 8¹³⁹ [i]