MinT - Best Known (163−101, 163, s)-Nets in Base 8

Best Known (163−101, 163, s)-Nets in Base 8

(163−101, 163, 98)-Net over F₈ — Constructive and digital

Digital (62, 163, 98)-net over F₈, using

t-expansion [i] based on digital (37, 163, 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]

(163−101, 163, 144)-Net over F₈ — Digital

Digital (62, 163, 144)-net over F₈, using

t-expansion [i] based on digital (45, 163, 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]

(163−101, 163, 2315)-Net in Base 8 — Upper bound on s

There is no (62, 163, 2316)-net in base 8, because

1 times m-reduction [i] would yield (62, 162, 2316)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 200 556653 072199 081563 033533 195197 975027 171060 904235 295311 483127 877428 716424 850497 648083 711630 959636 404438 487066 684305 931408 132235 418815 904029 802192 > 8¹⁶² [i]