MinT - Best Known (133−47, 133, s)-Nets in Base 8

Best Known (133−47, 133, s)-Nets in Base 8

(133−47, 133, 371)-Net over F₈ — Constructive and digital

Digital (86, 133, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 25, 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 (61, 108, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 54, 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]

(133−47, 133, 576)-Net in Base 8 — Constructive

(86, 133, 576)-net in base 8, using

1 times m-reduction [i] based on (86, 134, 576)-net in base 8, using
- trace code for nets [i] based on (19, 67, 288)-net in base 64, using
  - 3 times m-reduction [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]

(133−47, 133, 1073)-Net over F₈ — Digital

Digital (86, 133, 1073)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(133−47, 133, 205248)-Net in Base 8 — Upper bound on s

There is no (86, 133, 205249)-net in base 8, because

1 times m-reduction [i] would yield (86, 132, 205249)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 161406 060138 283075 234166 763077 516218 190494 954141 179782 350270 626779 168674 802727 397345 173921 607968 583790 617155 502894 203840 > 8¹³² [i]