MinT - Best Known (167−117, 167, s)-Nets in Base 8

Best Known (167−117, 167, s)-Nets in Base 8

(167−117, 167, 98)-Net over F₈ — Constructive and digital

Digital (50, 167, 98)-net over F₈, using

t-expansion [i] based on digital (37, 167, 98)-net over F₈, using
- net from sequence [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(167−117, 167, 144)-Net over F₈ — Digital

Digital (50, 167, 144)-net over F₈, using

t-expansion [i] based on digital (45, 167, 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]

(167−117, 167, 1196)-Net in Base 8 — Upper bound on s

There is no (50, 167, 1197)-net in base 8, because

1 times m-reduction [i] would yield (50, 166, 1197)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 850583 446893 798864 004978 154168 342367 367830 522897 533888 705950 601330 095056 189196 236601 050090 290548 812364 642015 966153 051040 876654 319843 995987 923352 577712 > 8¹⁶⁶ [i]