MinT - Best Known (85−36, 85, s)-Nets in Base 8

Best Known (85−36, 85, s)-Nets in Base 8

(85−36, 85, 256)-Net over F₈ — Constructive and digital

Digital (49, 85, 256)-net over F₈, using

3 times m-reduction [i] based on digital (49, 88, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 44, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(85−36, 85, 258)-Net in Base 8 — Constructive

(49, 85, 258)-net in base 8, using

8¹ times duplication [i] based on (48, 84, 258)-net in base 8, using
- trace code for nets [i] based on (6, 42, 129)-net in base 64, using
  - base change [i] based on digital (0, 36, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]

(85−36, 85, 322)-Net over F₈ — Digital

Digital (49, 85, 322)-net over F₈, using

1 times m-reduction [i] based on digital (49, 86, 322)-net over F₈, using
- trace code for nets [i] based on digital (6, 43, 161)-net over F₆₄, using
  - net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
      - a curve from the manYPoints database [i]

(85−36, 85, 19832)-Net in Base 8 — Upper bound on s

There is no (49, 85, 19833)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 57913 125783 219469 224132 014515 330130 372161 120508 647544 572045 427515 224134 645276 > 8⁸⁵ [i]