MinT - Best Known (158−57, 158, s)-Nets in Base 8

Best Known (158−57, 158, s)-Nets in Base 8

(158−57, 158, 371)-Net over F₈ — Constructive and digital

Digital (101, 158, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 30, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (71, 128, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 64, 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]

(158−57, 158, 576)-Net in Base 8 — Constructive

(101, 158, 576)-net in base 8, using

2 times m-reduction [i] based on (101, 160, 576)-net in base 8, using
- trace code for nets [i] based on (21, 80, 288)-net in base 64, using
  - 4 times m-reduction [i] based on (21, 84, 288)-net in base 64, using
    - base change [i] based on digital (9, 72, 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]

(158−57, 158, 1122)-Net over F₈ — Digital

Digital (101, 158, 1122)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(158−57, 158, 186904)-Net in Base 8 — Upper bound on s

There is no (101, 158, 186905)-net in base 8, because

1 times m-reduction [i] would yield (101, 157, 186905)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6097 173167 880548 926001 152647 941504 924168 988268 988822 982389 526808 097314 878163 529747 499573 277417 292010 529632 876833 743032 306988 927318 344710 812800 > 8¹⁵⁷ [i]