MinT - Best Known (73, 73+41, s)-Nets in Base 8

Best Known (73, 73+41, s)-Nets in Base 8

(73, 73+41, 354)-Net over F₈ — Constructive and digital

Digital (73, 114, 354)-net over F₈, using

18 times m-reduction [i] based on digital (73, 132, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 66, 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]

(73, 73+41, 516)-Net in Base 8 — Constructive

(73, 114, 516)-net in base 8, using

8² times duplication [i] based on (71, 112, 516)-net in base 8, using
- base change [i] based on digital (43, 84, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 42, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(73, 73+41, 864)-Net over F₈ — Digital

Digital (73, 114, 864)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(73, 73+41, 150186)-Net in Base 8 — Upper bound on s

There is no (73, 114, 150187)-net in base 8, because

1 times m-reduction [i] would yield (73, 113, 150187)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 119947 443355 452438 247463 922185 981307 577486 610692 278348 702976 562008 030734 045799 444438 793668 022132 266924 > 8¹¹³ [i]