MinT - Best Known (129−39, 129, s)-Nets in Base 8

Best Known (129−39, 129, s)-Nets in Base 8

(129−39, 129, 419)-Net over F₈ — Constructive and digital

Digital (90, 129, 419)-net over F₈, using

t-expansion [i] based on digital (89, 129, 419)-net over F₈, using
- 1 times m-reduction [i] based on digital (89, 130, 419)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (14, 34, 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 (55, 96, 354)-net over F₈, using
      - trace code for nets [i] based on digital (7, 48, 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]

(129−39, 129, 576)-Net in Base 8 — Constructive

(90, 129, 576)-net in base 8, using

t-expansion [i] based on (89, 129, 576)-net in base 8, using
- 11 times m-reduction [i] based on (89, 140, 576)-net in base 8, using
  - trace code for nets [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 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]

(129−39, 129, 2516)-Net over F₈ — Digital

Digital (90, 129, 2516)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(129−39, 129, 1374321)-Net in Base 8 — Upper bound on s

There is no (90, 129, 1374322)-net in base 8, because

1 times m-reduction [i] would yield (90, 128, 1374322)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 39 402152 512879 739090 897303 363191 251004 826919 448871 151988 227368 468539 096121 983919 545856 009054 189081 232548 506896 380912 > 8¹²⁸ [i]