MinT - Best Known (80−34, 80, s)-Nets in Base 8

Best Known (80−34, 80, s)-Nets in Base 8

(80−34, 80, 256)-Net over F₈ — Constructive and digital

Digital (46, 80, 256)-net over F₈, using

2 times m-reduction [i] based on digital (46, 82, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 41, 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]

(80−34, 80, 258)-Net in Base 8 — Constructive

(46, 80, 258)-net in base 8, using

trace code for nets [i] based on (6, 40, 129)-net in base 64, using
- 2 times m-reduction [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]

(80−34, 80, 322)-Net over F₈ — Digital

Digital (46, 80, 322)-net over F₈, using

trace code for nets [i] based on digital (6, 40, 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]

(80−34, 80, 18215)-Net in Base 8 — Upper bound on s

There is no (46, 80, 18216)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 768170 864656 682086 220157 066127 037142 810491 931801 264546 269104 892338 097630 > 8⁸⁰ [i]