MinT - Best Known (169−99, 169, s)-Nets in Base 8

Best Known (169−99, 169, s)-Nets in Base 8

(169−99, 169, 99)-Net over F₈ — Constructive and digital

Digital (70, 169, 99)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 56, 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, 113, 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]

(169−99, 169, 144)-Net over F₈ — Digital

Digital (70, 169, 144)-net over F₈, using

t-expansion [i] based on digital (45, 169, 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]

(169−99, 169, 156)-Net in Base 8

(70, 169, 156)-net in base 8, using

3 times m-reduction [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]

(169−99, 169, 3377)-Net in Base 8 — Upper bound on s

There is no (70, 169, 3378)-net in base 8, because

1 times m-reduction [i] would yield (70, 168, 3378)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 52 952868 919562 348492 887944 885891 473495 654843 131613 233804 899028 988737 519523 325717 424174 212380 064650 996848 821209 388426 440556 929153 566368 140827 498530 779207 > 8¹⁶⁸ [i]