MinT - Best Known (138−52, 138, s)-Nets in Base 8

Best Known (138−52, 138, s)-Nets in Base 8

(138−52, 138, 354)-Net over F₈ — Constructive and digital

Digital (86, 138, 354)-net over F₈, using

20 times m-reduction [i] based on digital (86, 158, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 79, 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]

(138−52, 138, 432)-Net in Base 8 — Constructive

(86, 138, 432)-net in base 8, using

t-expansion [i] based on (85, 138, 432)-net in base 8, using
- 2 times m-reduction [i] based on (85, 140, 432)-net in base 8, using
  - trace code for nets [i] based on (15, 70, 216)-net in base 64, using
    - base change [i] based on digital (5, 60, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(138−52, 138, 802)-Net over F₈ — Digital

Digital (86, 138, 802)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−52, 138, 93636)-Net in Base 8 — Upper bound on s

There is no (86, 138, 93637)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 42317 478555 578216 613611 479253 265927 142111 151911 900166 716939 037261 247624 304019 359320 632461 026669 996145 010319 749685 897542 700880 > 8¹³⁸ [i]