MinT - Best Known (64−21, 64, s)-Nets in Base 8

Best Known (64−21, 64, s)-Nets in Base 8

(64−21, 64, 354)-Net over F₈ — Constructive and digital

Digital (43, 64, 354)-net over F₈, using

8 times m-reduction [i] based on digital (43, 72, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 36, 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]

(64−21, 64, 520)-Net in Base 8 — Constructive

(43, 64, 520)-net in base 8, using

base change [i] based on digital (27, 48, 520)-net over F₁₆, using
- trace code for nets [i] based on digital (3, 24, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(64−21, 64, 931)-Net over F₈ — Digital

Digital (43, 64, 931)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(64−21, 64, 316473)-Net in Base 8 — Upper bound on s

There is no (43, 64, 316474)-net in base 8, because

1 times m-reduction [i] would yield (43, 63, 316474)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 784 655612 907239 672238 284273 058531 538102 820111 330717 678776 > 8⁶³ [i]