MinT - Best Known (163−113, 163, s)-Nets in Base 8

Best Known (163−113, 163, s)-Nets in Base 8

(163−113, 163, 98)-Net over F₈ — Constructive and digital

Digital (50, 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]

(163−113, 163, 144)-Net over F₈ — Digital

Digital (50, 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]

(163−113, 163, 1235)-Net in Base 8 — Upper bound on s

There is no (50, 163, 1236)-net in base 8, because

1 times m-reduction [i] would yield (50, 162, 1236)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 204 012039 954923 972702 363606 268995 411284 158882 663599 186699 695920 257802 493253 408055 360883 569819 468740 017954 938283 018205 056204 427836 248207 613836 600376 > 8¹⁶² [i]