MinT - Best Known (64, 64+41, s)-Nets in Base 8

Best Known (64, 64+41, s)-Nets in Base 8

(64, 64+41, 354)-Net over F₈ — Constructive and digital

Digital (64, 105, 354)-net over F₈, using

9 times m-reduction [i] based on 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]

(64, 64+41, 384)-Net in Base 8 — Constructive

(64, 105, 384)-net in base 8, using

1 times m-reduction [i] based on (64, 106, 384)-net in base 8, using
- trace code for nets [i] based on (11, 53, 192)-net in base 64, using
  - 3 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]

(64, 64+41, 542)-Net over F₈ — Digital

Digital (64, 105, 542)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 64+41, 58909)-Net in Base 8 — Upper bound on s

There is no (64, 105, 58910)-net in base 8, because

1 times m-reduction [i] would yield (64, 104, 58910)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8345 229926 746745 253358 710570 496802 540801 252783 910666 406887 974756 064759 067152 048229 787302 051037 > 8¹⁰⁴ [i]