MinT - Best Known (82, 164, s)-Nets in Base 8

Best Known (82, 164, s)-Nets in Base 8

(82, 164, 130)-Net over F₈ — Constructive and digital

Digital (82, 164, 130)-net over F₈, using

trace code for nets [i] based on digital (0, 82, 65)-net over F₆₄, using
- net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
    - the rational function field F₆₄(x) [i]
  - Niederreiter sequence [i]

(82, 164, 225)-Net in Base 8 — Constructive

(82, 164, 225)-net in base 8, using

4 times m-reduction [i] based on (82, 168, 225)-net in base 8, using
- base change [i] based on digital (40, 126, 225)-net over F₁₆, using
  - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
      - a function field by GarcÃa and Quoos [i]

(82, 164, 263)-Net over F₈ — Digital

Digital (82, 164, 263)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 164, 9418)-Net in Base 8 — Upper bound on s

There is no (82, 164, 9419)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12810 355972 067586 920518 494554 609589 053190 735479 427209 792413 432280 112694 744999 065455 164511 189185 643643 140738 670808 562289 509262 966679 747651 895047 462380 > 8¹⁶⁴ [i]