MinT - Best Known (158−58, 158, s)-Nets in Base 8

Best Known (158−58, 158, s)-Nets in Base 8

(158−58, 158, 354)-Net over F₈ — Constructive and digital

Digital (100, 158, 354)-net over F₈, using

t-expansion [i] based on digital (93, 158, 354)-net over F₈, using
- 14 times m-reduction [i] based on digital (93, 172, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 86, 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]

(158−58, 158, 576)-Net in Base 8 — Constructive

(100, 158, 576)-net in base 8, using

trace code for nets [i] based on (21, 79, 288)-net in base 64, using
- 5 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]

(158−58, 158, 1024)-Net over F₈ — Digital

Digital (100, 158, 1024)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(158−58, 158, 138755)-Net in Base 8 — Upper bound on s

There is no (100, 158, 138756)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 48781 182944 422753 450363 143840 484719 336994 899876 978037 645300 433836 318223 131433 434321 396383 812005 191901 641310 712872 740579 854447 338894 827058 560128 > 8¹⁵⁸ [i]