MinT - Best Known (172−157, 172, s)-Nets in Base 4

Best Known (172−157, 172, s)-Nets in Base 4

(172−157, 172, 33)-Net over F₄ — Constructive and digital

Digital (15, 172, 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]

(172−157, 172, 35)-Net over F₄ — Digital

Digital (15, 172, 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]

(172−157, 172, 56)-Net in Base 4 — Upper bound on s

There is no (15, 172, 57)-net in base 4, because

5 times m-reduction [i] would yield (15, 167, 57)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁶⁷, 57, S₄, 3, 152), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 699920 231930 563815 799200 712677 638836 913014 217885 827979 656246 594051 184959 743800 295431 524216 647377 223680 / 17 > 4¹⁶⁷ [i]