MinT - Best Known (82, 82+83, s)-Nets in Base 8

Best Known (82, 82+83, s)-Nets in Base 8

(82, 82+83, 130)-Net over F₈ — Constructive and digital

Digital (82, 165, 130)-net over F₈, using

t-expansion [i] based on digital (76, 165, 130)-net over F₈, using
- 7 times m-reduction [i] based on digital (76, 172, 130)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (14, 62, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
    2. digital (14, 110, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(82, 82+83, 225)-Net in Base 8 — Constructive

(82, 165, 225)-net in base 8, using

3 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, 82+83, 258)-Net over F₈ — Digital

Digital (82, 165, 258)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+83, 9418)-Net in Base 8 — Upper bound on s

There is no (82, 165, 9419)-net in base 8, because

1 times m-reduction [i] would yield (82, 164, 9419)-net in base 8, but
- 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]