MinT - Best Known (157−57, 157, s)-Nets in Base 8

Best Known (157−57, 157, s)-Nets in Base 8

(157−57, 157, 368)-Net over F₈ — Constructive and digital

Digital (100, 157, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 29, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (71, 128, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 64, 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]

(157−57, 157, 576)-Net in Base 8 — Constructive

(100, 157, 576)-net in base 8, using

1 times m-reduction [i] based on (100, 158, 576)-net in base 8, using
- trace code for nets [i] based on (21, 79, 288)-net in base 64, using
  - 5 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]

(157−57, 157, 1080)-Net over F₈ — Digital

Digital (100, 157, 1080)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−57, 157, 173525)-Net in Base 8 — Upper bound on s

There is no (100, 157, 173526)-net in base 8, because

1 times m-reduction [i] would yield (100, 156, 173526)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 762 145896 685760 731288 111557 164880 767347 912358 890075 705467 786933 495854 876501 983630 914674 984481 717878 436835 075385 359884 686740 650634 774220 285960 > 8¹⁵⁶ [i]