MinT - Best Known (107−35, 107, s)-Nets in Base 8

Best Known (107−35, 107, s)-Nets in Base 8

(107−35, 107, 382)-Net over F₈ — Constructive and digital

Digital (72, 107, 382)-net over F₈, using

8¹ times duplication [i] based on digital (71, 106, 382)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 22, 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 (49, 84, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 42, 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]

(107−35, 107, 576)-Net in Base 8 — Constructive

(72, 107, 576)-net in base 8, using

3 times m-reduction [i] based on (72, 110, 576)-net in base 8, using
- trace code for nets [i] based on (17, 55, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
    - base change [i] based on digital (9, 48, 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]

(107−35, 107, 1361)-Net over F₈ — Digital

Digital (72, 107, 1361)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(107−35, 107, 438397)-Net in Base 8 — Upper bound on s

There is no (72, 107, 438398)-net in base 8, because

1 times m-reduction [i] would yield (72, 106, 438398)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 533997 691284 219877 161602 584532 248597 242925 814202 672001 291571 052681 899139 987420 581704 187255 730707 > 8¹⁰⁶ [i]