MinT - Best Known (65, 65+43, s)-Nets in Base 8

Best Known (65, 65+43, s)-Nets in Base 8

(65, 65+43, 354)-Net over F₈ — Constructive and digital

Digital (65, 108, 354)-net over F₈, using

8 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+43, 384)-Net in Base 8 — Constructive

(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+43, 503)-Net over F₈ — Digital

Digital (65, 108, 503)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+43, 49515)-Net in Base 8 — Upper bound on s

There is no (65, 108, 49516)-net in base 8, because

1 times m-reduction [i] would yield (65, 107, 49516)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 273320 677952 797189 623512 008681 627823 929747 416495 957053 705054 827706 587074 920076 385329 951399 114324 > 8¹⁰⁷ [i]