MinT - Best Known (167−113, 167, s)-Nets in Base 8

Best Known (167−113, 167, s)-Nets in Base 8

(167−113, 167, 98)-Net over F₈ — Constructive and digital

Digital (54, 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]

(167−113, 167, 144)-Net over F₈ — Digital

Digital (54, 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]

(167−113, 167, 1438)-Net in Base 8 — Upper bound on s

There is no (54, 167, 1439)-net in base 8, because

1 times m-reduction [i] would yield (54, 166, 1439)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 819402 701277 424626 268043 751373 974106 910357 137575 350609 767974 333315 109730 755701 598735 000945 089036 064251 803238 962737 253111 133352 299287 174667 530682 160676 > 8¹⁶⁶ [i]