MinT - Best Known (128−40, 128, s)-Nets in Base 8

Best Known (128−40, 128, s)-Nets in Base 8

(128−40, 128, 419)-Net over F₈ — Constructive and digital

Digital (88, 128, 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 (54, 94, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 47, 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]

(128−40, 128, 576)-Net in Base 8 — Constructive

(88, 128, 576)-net in base 8, using

10 times m-reduction [i] based on (88, 138, 576)-net in base 8, using
- trace code for nets [i] based on (19, 69, 288)-net in base 64, using
  - 1 times m-reduction [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]

(128−40, 128, 2044)-Net over F₈ — Digital

Digital (88, 128, 2044)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−40, 128, 714457)-Net in Base 8 — Upper bound on s

There is no (88, 128, 714458)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 39 402200 930294 952980 748191 471281 452713 085117 360093 472813 411252 370034 393045 352808 892951 910351 507632 111347 768444 468482 > 8¹²⁸ [i]