MinT - Best Known (142−49, 142, s)-Nets in Base 8

Best Known (142−49, 142, s)-Nets in Base 8

(142−49, 142, 382)-Net over F₈ — Constructive and digital

Digital (93, 142, 382)-net over F₈, using

8¹ times duplication [i] based on digital (92, 141, 382)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 29, 28)-net over F₈, using
    - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
        a function field by SÃ©mirat [i]
  2. digital (63, 112, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 56, 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]

(142−49, 142, 576)-Net in Base 8 — Constructive

(93, 142, 576)-net in base 8, using

4 times m-reduction [i] based on (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]

(142−49, 142, 1281)-Net over F₈ — Digital

Digital (93, 142, 1281)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(142−49, 142, 283071)-Net in Base 8 — Upper bound on s

There is no (93, 142, 283072)-net in base 8, because

1 times m-reduction [i] would yield (93, 141, 283072)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 21 663213 305749 270230 604679 641340 914952 202566 144483 052838 506128 891243 730798 255186 243468 011508 084462 915002 665575 299158 657960 840277 > 8¹⁴¹ [i]