MinT - Best Known (133−42, 133, s)-Nets in Base 8

Best Known (133−42, 133, s)-Nets in Base 8

(133−42, 133, 419)-Net over F₈ — Constructive and digital

Digital (91, 133, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 35, 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 (56, 98, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 49, 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]

(133−42, 133, 576)-Net in Base 8 — Constructive

(91, 133, 576)-net in base 8, using

t-expansion [i] based on (89, 133, 576)-net in base 8, using
- 7 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]

(133−42, 133, 1981)-Net over F₈ — Digital

Digital (91, 133, 1981)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(133−42, 133, 650063)-Net in Base 8 — Upper bound on s

There is no (91, 133, 650064)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 291147 159406 243722 029778 422961 071053 768857 517564 979471 266027 461303 934660 793076 476139 224724 894837 205304 466783 052040 480676 > 8¹³³ [i]