MinT - Best Known (138−47, 138, s)-Nets in Base 8

Best Known (138−47, 138, s)-Nets in Base 8

(138−47, 138, 388)-Net over F₈ — Constructive and digital

Digital (91, 138, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 30, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (61, 108, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 54, 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]

(138−47, 138, 576)-Net in Base 8 — Constructive

(91, 138, 576)-net in base 8, using

t-expansion [i] based on (89, 138, 576)-net in base 8, using
- 2 times m-reduction [i] based on (89, 140, 576)-net in base 8, using
  - trace code for nets [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 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]

(138−47, 138, 1339)-Net over F₈ — Digital

Digital (91, 138, 1339)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−47, 138, 322562)-Net in Base 8 — Upper bound on s

There is no (91, 138, 322563)-net in base 8, because

1 times m-reduction [i] would yield (91, 137, 322563)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5288 560794 438410 336785 793252 114587 094278 860948 171361 345644 728972 027973 642736 502819 379323 112722 220712 119138 946823 652738 612224 > 8¹³⁷ [i]