MinT - Best Known (212−190, 212, s)-Nets in Base 4

Best Known (212−190, 212, s)-Nets in Base 4

(212−190, 212, 34)-Net over F₄ — Constructive and digital

Digital (22, 212, 34)-net over F₄, using

t-expansion [i] based on digital (21, 212, 34)-net over F₄, using
- net from sequence [i] based on digital (21, 33)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
    - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(212−190, 212, 44)-Net over F₄ — Digital

Digital (22, 212, 44)-net over F₄, using

t-expansion [i] based on digital (21, 212, 44)-net over F₄, using
- net from sequence [i] based on digital (21, 43)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 44, using
    - a curve from the manYPoints database [i]

(212−190, 212, 81)-Net in Base 4 — Upper bound on s

There is no (22, 212, 82)-net in base 4, because

52 times m-reduction [i] would yield (22, 160, 82)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁶⁰, 82, S₄, 2, 138), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 341 757925 747345 613183 203472 987128 338336 432723 577064 443191 526657 251555 156124 902488 003673 933909 852160 / 139 > 4¹⁶⁰ [i]