MinT - Best Known (52, 52+111, s)-Nets in Base 8

Best Known (52, 52+111, s)-Nets in Base 8

(52, 52+111, 98)-Net over F₈ — Constructive and digital

Digital (52, 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]

(52, 52+111, 144)-Net over F₈ — Digital

Digital (52, 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]

(52, 52+111, 1358)-Net in Base 8 — Upper bound on s

There is no (52, 163, 1359)-net in base 8, because

1 times m-reduction [i] would yield (52, 162, 1359)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 201 480224 665465 478800 094590 992080 407659 346379 504045 948772 968078 649025 387580 508305 789692 722187 688661 820930 914956 114077 868716 668295 008510 849175 107520 > 8¹⁶² [i]