MinT - Best Known (135−45, 135, s)-Nets in Base 8

Best Known (135−45, 135, s)-Nets in Base 8

(135−45, 135, 399)-Net over F₈ — Constructive and digital

Digital (90, 135, 399)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 31, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]
2. digital (59, 104, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 52, 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]

(135−45, 135, 576)-Net in Base 8 — Constructive

(90, 135, 576)-net in base 8, using

t-expansion [i] based on (89, 135, 576)-net in base 8, using
- 5 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]

(135−45, 135, 1476)-Net over F₈ — Digital

Digital (90, 135, 1476)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−45, 135, 409618)-Net in Base 8 — Upper bound on s

There is no (90, 135, 409619)-net in base 8, because

1 times m-reduction [i] would yield (90, 134, 409619)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 10 329291 856756 153140 362784 188789 391181 455814 043697 288341 545873 940171 866472 669238 698180 124326 511128 150658 216596 400259 792896 > 8¹³⁴ [i]