MinT - Best Known (71, 172, s)-Nets in Base 8

Best Known (71, 172, s)-Nets in Base 8

(71, 172, 99)-Net over F₈ — Constructive and digital

Digital (71, 172, 99)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 57, 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 (14, 115, 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]

(71, 172, 144)-Net over F₈ — Digital

Digital (71, 172, 144)-net over F₈, using

t-expansion [i] based on digital (45, 172, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(71, 172, 156)-Net in Base 8

(71, 172, 156)-net in base 8, using

t-expansion [i] based on (70, 172, 156)-net in base 8, using
- base change [i] based on digital (27, 129, 156)-net over F₁₆, using
  - net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
      - a curve from the manYPoints database [i]

(71, 172, 3381)-Net in Base 8 — Upper bound on s

There is no (71, 172, 3382)-net in base 8, because

1 times m-reduction [i] would yield (71, 171, 3382)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 27049 966201 193738 737737 100636 752341 335205 187602 630081 055844 400811 353275 297800 679993 889957 746977 166321 625034 809019 592569 442175 858819 733257 498131 487327 462434 > 8¹⁷¹ [i]