MinT - Best Known (69−24, 69, s)-Nets in Base 8

Best Known (69−24, 69, s)-Nets in Base 8

(69−24, 69, 354)-Net over F₈ — Constructive and digital

Digital (45, 69, 354)-net over F₈, using

7 times m-reduction [i] based on digital (45, 76, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 38, 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]

(69−24, 69, 516)-Net in Base 8 — Constructive

(45, 69, 516)-net in base 8, using

1 times m-reduction [i] based on (45, 70, 516)-net in base 8, using
- trace code for nets [i] based on (10, 35, 258)-net in base 64, using
  - 1 times m-reduction [i] based on (10, 36, 258)-net in base 64, using
    - base change [i] based on digital (1, 27, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(69−24, 69, 701)-Net over F₈ — Digital

Digital (45, 69, 701)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−24, 69, 117761)-Net in Base 8 — Upper bound on s

There is no (45, 69, 117762)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 205 693629 320177 883599 535935 503107 926277 624390 148718 876933 075712 > 8⁶⁹ [i]