MinT - Best Known (137−122, 137, s)-Nets in Base 4

Best Known (137−122, 137, s)-Nets in Base 4

(137−122, 137, 33)-Net over F₄ — Constructive and digital

Digital (15, 137, 33)-net over F₄, using

net from sequence [i] based on digital (15, 32)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
  - an explicitly constructive algebraic function field [i]

(137−122, 137, 35)-Net over F₄ — Digital

Digital (15, 137, 35)-net over F₄, using

net from sequence [i] based on digital (15, 34)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 35, using
  - a curve from the manYPoints database [i]

(137−122, 137, 58)-Net in Base 4 — Upper bound on s

There is no (15, 137, 59)-net in base 4, because

23 times m-reduction [i] would yield (15, 114, 59)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹¹⁴, 59, S₄, 2, 99), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 12078 056106 883486 628010 822758 984794 541789 440701 298176 471534 417391 648768 / 25 > 4¹¹⁴ [i]