MinT - Best Known (54, 82, s)-Nets in Base 8

Best Known (54, 82, s)-Nets in Base 8

(54, 82, 354)-Net over F₈ — Constructive and digital

Digital (54, 82, 354)-net over F₈, using

12 times m-reduction [i] based on digital (54, 94, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 47, 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]

(54, 82, 518)-Net in Base 8 — Constructive

(54, 82, 518)-net in base 8, using

8² times duplication [i] based on (52, 80, 518)-net in base 8, using
- base change [i] based on digital (32, 60, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 30, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(54, 82, 876)-Net over F₈ — Digital

Digital (54, 82, 876)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 82, 168215)-Net in Base 8 — Upper bound on s

There is no (54, 82, 168216)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 113 087251 607537 824527 998220 191188 288853 379905 086530 980117 347302 223915 466502 > 8⁸² [i]