MinT - Best Known (127−39, 127, s)-Nets in Base 8

Best Known (127−39, 127, s)-Nets in Base 8

(127−39, 127, 419)-Net over F₈ — Constructive and digital

Digital (88, 127, 419)-net over F₈, using

1 times m-reduction [i] based on digital (88, 128, 419)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 34, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
  2. digital (54, 94, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 47, 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]

(127−39, 127, 576)-Net in Base 8 — Constructive

(88, 127, 576)-net in base 8, using

11 times m-reduction [i] based on (88, 138, 576)-net in base 8, using
- trace code for nets [i] based on (19, 69, 288)-net in base 64, using
  - 1 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]

(127−39, 127, 2257)-Net over F₈ — Digital

Digital (88, 127, 2257)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−39, 127, 1104143)-Net in Base 8 — Upper bound on s

There is no (88, 127, 1104144)-net in base 8, because

1 times m-reduction [i] would yield (88, 126, 1104144)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 615664 768770 846328 296789 185017 218283 268226 127263 242083 341435 157182 487243 696742 611687 200750 772892 913413 166690 380560 > 8¹²⁶ [i]