MinT - Best Known (70, 157, s)-Nets in Base 3

Best Known (70, 157, s)-Nets in Base 3

(70, 157, 48)-Net over F₃ — Constructive and digital

Digital (70, 157, 48)-net over F₃, using

t-expansion [i] based on digital (45, 157, 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, 157, 82)-Net over F₃ — Digital

Digital (70, 157, 82)-net over F₃, using

t-expansion [i] based on digital (69, 157, 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, 157, 413)-Net in Base 3 — Upper bound on s

There is no (70, 157, 414)-net in base 3, because

1 times m-reduction [i] would yield (70, 156, 414)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 284 783795 647867 269846 488369 006483 945116 074211 081970 993606 850972 161691 728625 > 3¹⁵⁶ [i]