MinT - Best Known (146−53, 146, s)-Nets in Base 8

Best Known (146−53, 146, s)-Nets in Base 8

(146−53, 146, 363)-Net over F₈ — Constructive and digital

Digital (93, 146, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 26, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (67, 120, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 60, 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]

(146−53, 146, 576)-Net in Base 8 — Constructive

(93, 146, 576)-net in base 8, using

trace code for nets [i] based on (20, 73, 288)-net in base 64, using
- 4 times m-reduction [i] based on (20, 77, 288)-net in base 64, using
  - base change [i] based on digital (9, 66, 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]

(146−53, 146, 1018)-Net over F₈ — Digital

Digital (93, 146, 1018)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(146−53, 146, 163913)-Net in Base 8 — Upper bound on s

There is no (93, 146, 163914)-net in base 8, because

1 times m-reduction [i] would yield (93, 145, 163914)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 88728 706534 945021 335894 372260 484844 153401 000394 426092 508330 953459 991408 843843 386041 514646 305450 448632 463301 590576 419649 800518 852000 > 8¹⁴⁵ [i]