MinT - Best Known (132−43, 132, s)-Nets in Base 8

Best Known (132−43, 132, s)-Nets in Base 8

(132−43, 132, 402)-Net over F₈ — Constructive and digital

Digital (89, 132, 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 (57, 100, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 50, 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]

(132−43, 132, 576)-Net in Base 8 — Constructive

(89, 132, 576)-net in base 8, using

8 times m-reduction [i] based on (89, 140, 576)-net in base 8, using
- trace code for nets [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]

(132−43, 132, 1646)-Net over F₈ — Digital

Digital (89, 132, 1646)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(132−43, 132, 533267)-Net in Base 8 — Upper bound on s

There is no (89, 132, 533268)-net in base 8, because

1 times m-reduction [i] would yield (89, 131, 533268)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 20174 057377 702732 025914 573029 904835 010549 326984 192172 545512 797683 716473 184827 186061 259882 050218 994901 430691 444077 025856 > 8¹³¹ [i]