MinT - Best Known (83−29, 83, s)-Nets in Base 8

Best Known (83−29, 83, s)-Nets in Base 8

(83−29, 83, 354)-Net over F₈ — Constructive and digital

Digital (54, 83, 354)-net over F₈, using

11 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]

(83−29, 83, 516)-Net in Base 8 — Constructive

(54, 83, 516)-net in base 8, using

1 times m-reduction [i] based on (54, 84, 516)-net in base 8, using
- base change [i] based on digital (33, 63, 516)-net over F₁₆, using
  - 1 times m-reduction [i] based on digital (33, 64, 516)-net over F₁₆, using
    - trace code for nets [i] based on digital (1, 32, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(83−29, 83, 781)-Net over F₈ — Digital

Digital (54, 83, 781)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(83−29, 83, 168215)-Net in Base 8 — Upper bound on s

There is no (54, 83, 168216)-net in base 8, because

1 times m-reduction [i] would yield (54, 82, 168216)-net in base 8, but
- 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]