MinT - Best Known (137−44, 137, s)-Nets in Base 8

Best Known (137−44, 137, s)-Nets in Base 8

(137−44, 137, 402)-Net over F₈ — Constructive and digital

Digital (93, 137, 402)-net over F₈, using

t-expansion [i] based on digital (92, 137, 402)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 33, 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 (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]

(137−44, 137, 576)-Net in Base 8 — Constructive

(93, 137, 576)-net in base 8, using

9 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]

(137−44, 137, 1840)-Net over F₈ — Digital

Digital (93, 137, 1840)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−44, 137, 543915)-Net in Base 8 — Upper bound on s

There is no (93, 137, 543916)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5288 472985 467657 037081 948726 023842 008159 907680 697671 385100 471459 205923 163298 572843 296640 027915 746056 359958 740076 517411 162772 > 8¹³⁷ [i]