MinT - Best Known (157−103, 157, s)-Nets in Base 8

Best Known (157−103, 157, s)-Nets in Base 8

(157−103, 157, 98)-Net over F₈ — Constructive and digital

Digital (54, 157, 98)-net over F₈, using

t-expansion [i] based on digital (37, 157, 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]

(157−103, 157, 144)-Net over F₈ — Digital

Digital (54, 157, 144)-net over F₈, using

t-expansion [i] based on digital (45, 157, 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]

(157−103, 157, 1609)-Net in Base 8 — Upper bound on s

There is no (54, 157, 1610)-net in base 8, because

1 times m-reduction [i] would yield (54, 156, 1610)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 784 623618 829930 760969 224447 308815 242422 218837 546129 255196 266955 979223 352130 936528 554801 811359 241721 256970 019266 469911 786315 764210 751847 280656 > 8¹⁵⁶ [i]