MinT - Best Known (85, 85+49, s)-Nets in Base 8

Best Known (85, 85+49, s)-Nets in Base 8

(85, 85+49, 354)-Net over F₈ — Constructive and digital

Digital (85, 134, 354)-net over F₈, using

22 times m-reduction [i] based on digital (85, 156, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 78, 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]

(85, 85+49, 516)-Net in Base 8 — Constructive

(85, 134, 516)-net in base 8, using

trace code for nets [i] based on (18, 67, 258)-net in base 64, using
- 1 times m-reduction [i] based on (18, 68, 258)-net in base 64, using
  - base change [i] based on digital (1, 51, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(85, 85+49, 913)-Net over F₈ — Digital

Digital (85, 134, 913)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+49, 141528)-Net in Base 8 — Upper bound on s

There is no (85, 134, 141529)-net in base 8, because

1 times m-reduction [i] would yield (85, 133, 141529)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 291329 708093 495164 354895 405777 300237 784136 710860 124414 140145 270464 429482 220440 474239 558169 854183 081159 219109 047269 097373 > 8¹³³ [i]