MinT - Best Known (168−84, 168, s)-Nets in Base 8

Best Known (168−84, 168, s)-Nets in Base 8

(168−84, 168, 130)-Net over F₈ — Constructive and digital

Digital (84, 168, 130)-net over F₈, using

trace code for nets [i] based on digital (0, 84, 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]

(168−84, 168, 225)-Net in Base 8 — Constructive

(84, 168, 225)-net in base 8, using

t-expansion [i] based on (83, 168, 225)-net in base 8, using
- 4 times m-reduction [i] based on (83, 172, 225)-net in base 8, using
  - base change [i] based on digital (40, 129, 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]

(168−84, 168, 269)-Net over F₈ — Digital

Digital (84, 168, 269)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−84, 168, 9635)-Net in Base 8 — Upper bound on s

There is no (84, 168, 9636)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 52 499720 345098 685181 759458 879038 643346 787673 228451 603563 499090 557771 296575 189399 942213 169670 529628 652601 766619 657731 525114 011686 992854 336406 166789 051312 > 8¹⁶⁸ [i]