MinT - Best Known (129−41, 129, s)-Nets in Base 8

Best Known (129−41, 129, s)-Nets in Base 8

(129−41, 129, 402)-Net over F₈ — Constructive and digital

Digital (88, 129, 402)-net over F₈, using

1 times m-reduction [i] based on digital (88, 130, 402)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 32, 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 (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]

(129−41, 129, 576)-Net in Base 8 — Constructive

(88, 129, 576)-net in base 8, using

9 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]

(129−41, 129, 1861)-Net over F₈ — Digital

Digital (88, 129, 1861)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(129−41, 129, 714457)-Net in Base 8 — Upper bound on s

There is no (88, 129, 714458)-net in base 8, because

1 times m-reduction [i] would yield (88, 128, 714458)-net in base 8, but
- 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]