MinT - Best Known (131−41, 131, s)-Nets in Base 8

Best Known (131−41, 131, s)-Nets in Base 8

(131−41, 131, 419)-Net over F₈ — Constructive and digital

Digital (90, 131, 419)-net over F₈, using

8¹ times duplication [i] based on digital (89, 130, 419)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 34, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
  2. digital (55, 96, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 48, 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]

(131−41, 131, 576)-Net in Base 8 — Constructive

(90, 131, 576)-net in base 8, using

t-expansion [i] based on (89, 131, 576)-net in base 8, using
- 9 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]

(131−41, 131, 2063)-Net over F₈ — Digital

Digital (90, 131, 2063)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(131−41, 131, 879603)-Net in Base 8 — Upper bound on s

There is no (90, 131, 879604)-net in base 8, because

1 times m-reduction [i] would yield (90, 130, 879604)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2521 750519 390140 435714 145292 418190 019620 060998 111906 595236 716311 194488 007941 052300 746187 770867 190592 699487 485084 771816 > 8¹³⁰ [i]