MinT - Best Known (68, 105, s)-Nets in Base 8

Best Known (68, 105, s)-Nets in Base 8

(68, 105, 354)-Net over F₈ — Constructive and digital

Digital (68, 105, 354)-net over F₈, using

17 times m-reduction [i] based on digital (68, 122, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 61, 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]

(68, 105, 518)-Net in Base 8 — Constructive

(68, 105, 518)-net in base 8, using

8¹ times duplication [i] based on (67, 104, 518)-net in base 8, using
- base change [i] based on digital (41, 78, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 39, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(68, 105, 896)-Net over F₈ — Digital

Digital (68, 105, 896)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 105, 178179)-Net in Base 8 — Upper bound on s

There is no (68, 105, 178180)-net in base 8, because

1 times m-reduction [i] would yield (68, 104, 178180)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8343 928916 689973 534263 205211 687470 201480 373162 599830 496708 164170 454415 493317 947272 623964 356992 > 8¹⁰⁴ [i]