MinT - Best Known (70, 70+91, s)-Nets in Base 3

Best Known (70, 70+91, s)-Nets in Base 3

(70, 70+91, 48)-Net over F₃ — Constructive and digital

Digital (70, 161, 48)-net over F₃, using

t-expansion [i] based on digital (45, 161, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(70, 70+91, 82)-Net over F₃ — Digital

Digital (70, 161, 82)-net over F₃, using

t-expansion [i] based on digital (69, 161, 82)-net over F₃, using
- net from sequence [i] based on digital (69, 81)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 69 and N(F) ≥ 82, using
    - Drinfeld modules of rank 1 [i]

(70, 70+91, 395)-Net in Base 3 — Upper bound on s

There is no (70, 161, 396)-net in base 3, because

1 times m-reduction [i] would yield (70, 160, 396)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 23269 895777 784173 050108 514166 388721 131051 473935 446478 301097 227760 562688 221433 > 3¹⁶⁰ [i]