MinT - Best Known (68, 68+38, s)-Nets in Base 8

Best Known (68, 68+38, s)-Nets in Base 8

(68, 68+38, 354)-Net over F₈ — Constructive and digital

Digital (68, 106, 354)-net over F₈, using

16 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, 68+38, 516)-Net in Base 8 — Constructive

(68, 106, 516)-net in base 8, using

8² times duplication [i] based on (66, 104, 516)-net in base 8, using
- base change [i] based on digital (40, 78, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 39, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(68, 68+38, 828)-Net over F₈ — Digital

Digital (68, 106, 828)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+38, 123699)-Net in Base 8 — Upper bound on s

There is no (68, 106, 123700)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 534040 004607 187876 908122 053309 514797 930352 327599 860714 919130 312865 363213 282993 709880 928787 606321 > 8¹⁰⁶ [i]