MinT - Best Known (135−123, 135, s)-Nets in Base 4

Best Known (135−123, 135, s)-Nets in Base 4

(135−123, 135, 28)-Net over F₄ — Constructive and digital

Digital (12, 135, 28)-net over F₄, using

net from sequence [i] based on digital (12, 27)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
  - an explicitly constructive algebraic function field [i]

(135−123, 135, 29)-Net over F₄ — Digital

Digital (12, 135, 29)-net over F₄, using

net from sequence [i] based on digital (12, 28)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 29, using
  - a curve from the manYPoints database [i]

(135−123, 135, 49)-Net in Base 4 — Upper bound on s

There is no (12, 135, 50)-net in base 4, because

40 times m-reduction [i] would yield (12, 95, 50)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4⁹⁵, 50, S₄, 2, 83), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 14908 116621 543366 814109 999880 118207 738243 094180 602081 968128 / 7 > 4⁹⁵ [i]