MinT - Best Known (27, 27+19, s)-Nets in Base 8

Best Known (27, 27+19, s)-Nets in Base 8

(27, 27+19, 208)-Net over F₈ — Constructive and digital

Digital (27, 46, 208)-net over F₈, using

2 times m-reduction [i] based on digital (27, 48, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 24, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(27, 27+19, 258)-Net in Base 8 — Constructive

(27, 46, 258)-net in base 8, using

trace code for nets [i] based on (4, 23, 129)-net in base 64, using
- 5 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
  - base change [i] based on digital (0, 24, 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]

(27, 27+19, 258)-Net over F₈ — Digital

Digital (27, 46, 258)-net over F₈, using

trace code for nets [i] based on digital (4, 23, 129)-net over F₆₄, using
- net from sequence [i] based on digital (4, 128)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 4 and N(F) ≥ 129, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(27, 27+19, 19408)-Net in Base 8 — Upper bound on s

There is no (27, 46, 19409)-net in base 8, because

1 times m-reduction [i] would yield (27, 45, 19409)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 43572 450199 862111 883255 829999 791570 634240 > 8⁴⁵ [i]