MinT - Best Known (152−49, 152, s)-Nets in Base 8

Best Known (152−49, 152, s)-Nets in Base 8

(152−49, 152, 419)-Net over F₈ — Constructive and digital

Digital (103, 152, 419)-net over F₈, using

1 times m-reduction [i] based on digital (103, 153, 419)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 39, 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 (64, 114, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 57, 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]

(152−49, 152, 576)-Net in Base 8 — Constructive

(103, 152, 576)-net in base 8, using

12 times m-reduction [i] based on (103, 164, 576)-net in base 8, using
- trace code for nets [i] based on (21, 82, 288)-net in base 64, using
  - 2 times m-reduction [i] based on (21, 84, 288)-net in base 64, using
    - base change [i] based on digital (9, 72, 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]

(152−49, 152, 1962)-Net over F₈ — Digital

Digital (103, 152, 1962)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(152−49, 152, 673280)-Net in Base 8 — Upper bound on s

There is no (103, 152, 673281)-net in base 8, because

1 times m-reduction [i] would yield (103, 151, 673281)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 23258 998334 921227 537330 731477 650510 837527 058479 216928 359069 557933 047133 404292 561021 547102 242060 967261 112128 846463 467104 705831 887518 171072 > 8¹⁵¹ [i]