MinT - Best Known (130−45, 130, s)-Nets in Base 8

Best Known (130−45, 130, s)-Nets in Base 8

(130−45, 130, 379)-Net over F₈ — Constructive and digital

Digital (85, 130, 379)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 26, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (59, 104, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 52, 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]

(130−45, 130, 576)-Net in Base 8 — Constructive

(85, 130, 576)-net in base 8, using

2 times m-reduction [i] based on (85, 132, 576)-net in base 8, using
- trace code for nets [i] based on (19, 66, 288)-net in base 64, using
  - 4 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]

(130−45, 130, 1170)-Net over F₈ — Digital

Digital (85, 130, 1170)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(130−45, 130, 255342)-Net in Base 8 — Upper bound on s

There is no (85, 130, 255343)-net in base 8, because

1 times m-reduction [i] would yield (85, 129, 255343)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 315 230188 303177 179846 213268 751534 645916 665865 539003 907690 433848 588609 419909 661033 478429 671265 560919 897995 377329 622106 > 8¹²⁹ [i]