MinT - Best Known (107, 107+59, s)-Nets in Base 8

Best Known (107, 107+59, s)-Nets in Base 8

(107, 107+59, 382)-Net over F₈ — Constructive and digital

Digital (107, 166, 382)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 34, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (73, 132, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 66, 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]

(107, 107+59, 576)-Net in Base 8 — Constructive

(107, 166, 576)-net in base 8, using

t-expansion [i] based on (105, 166, 576)-net in base 8, using
- 2 times m-reduction [i] based on (105, 168, 576)-net in base 8, using
  - trace code for nets [i] based on (21, 84, 288)-net in base 64, using
    - base change [i] based on digital (9, 72, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(107, 107+59, 1262)-Net over F₈ — Digital

Digital (107, 166, 1262)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(107, 107+59, 229224)-Net in Base 8 — Upper bound on s

There is no (107, 166, 229225)-net in base 8, because

1 times m-reduction [i] would yield (107, 165, 229225)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102305 404866 615608 432200 219491 353024 685436 354993 148865 869790 922826 238799 688299 593706 179704 609720 571001 528708 305472 535916 205262 594437 362987 579167 524384 > 8¹⁶⁵ [i]