MinT - Best Known (168−72, 168, s)-Nets in Base 8

Best Known (168−72, 168, s)-Nets in Base 8

(168−72, 168, 354)-Net over F₈ — Constructive and digital

Digital (96, 168, 354)-net over F₈, using

t-expansion [i] based on digital (93, 168, 354)-net over F₈, using
- 4 times m-reduction [i] based on digital (93, 172, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 86, 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]

(168−72, 168, 514)-Net over F₈ — Digital

Digital (96, 168, 514)-net over F₈, using

trace code for nets [i] based on digital (12, 84, 257)-net over F₆₄, using
- net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(168−72, 168, 33401)-Net in Base 8 — Upper bound on s

There is no (96, 168, 33402)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 52 407810 495354 203843 651288 913470 488211 543674 860557 578599 522258 432353 973219 568241 771365 700718 812840 998736 976446 759656 283032 332205 190177 187307 140899 417250 > 8¹⁶⁸ [i]