MinT - Best Known (138−93, 138, s)-Nets in Base 8

Best Known (138−93, 138, s)-Nets in Base 8

(138−93, 138, 98)-Net over F₈ — Constructive and digital

Digital (45, 138, 98)-net over F₈, using

t-expansion [i] based on digital (37, 138, 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]

(138−93, 138, 144)-Net over F₈ — Digital

Digital (45, 138, 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]

(138−93, 138, 1229)-Net in Base 8 — Upper bound on s

There is no (45, 138, 1230)-net in base 8, because

1 times m-reduction [i] would yield (45, 137, 1230)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5408 135297 901160 491103 294001 424894 816227 787483 857689 117459 581573 977274 132111 932185 145509 247035 276187 904171 751093 768743 041536 > 8¹³⁷ [i]