MinT - Best Known (65, 65+41, s)-Nets in Base 8

Best Known (65, 65+41, s)-Nets in Base 8

(65, 65+41, 354)-Net over F₈ — Constructive and digital

Digital (65, 106, 354)-net over F₈, using

10 times m-reduction [i] based on digital (65, 116, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 58, 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]

(65, 65+41, 384)-Net in Base 8 — Constructive

(65, 106, 384)-net in base 8, using

2 times m-reduction [i] based on (65, 108, 384)-net in base 8, using
- trace code for nets [i] based on (11, 54, 192)-net in base 64, using
  - 2 times m-reduction [i] based on (11, 56, 192)-net in base 64, using
    - base change [i] based on digital (3, 48, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(65, 65+41, 573)-Net over F₈ — Digital

Digital (65, 106, 573)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+41, 65365)-Net in Base 8 — Upper bound on s

There is no (65, 106, 65366)-net in base 8, because

1 times m-reduction [i] would yield (65, 105, 65366)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 66759 972825 977381 793681 499678 788690 309884 891571 921072 202316 487081 863165 777541 197238 992684 993873 > 8¹⁰⁵ [i]