MinT - Best Known (130−41, 130, s)-Nets in Base 8

Best Known (130−41, 130, s)-Nets in Base 8

(130−41, 130, 419)-Net over F₈ — Constructive and digital

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]

(130−41, 130, 576)-Net in Base 8 — Constructive

(89, 130, 576)-net in base 8, using

10 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]

(130−41, 130, 1960)-Net over F₈ — Digital

Digital (89, 130, 1960)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(130−41, 130, 792741)-Net in Base 8 — Upper bound on s

There is no (89, 130, 792742)-net in base 8, because

1 times m-reduction [i] would yield (89, 129, 792742)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 315 216495 479906 974081 829230 870822 671351 842772 035278 278405 547030 135257 892009 416604 702444 736755 983200 408933 260924 474558 > 8¹²⁹ [i]