MinT - Best Known (172−136, 172, s)-Nets in Base 4

Best Known (172−136, 172, s)-Nets in Base 4

(172−136, 172, 56)-Net over F₄ — Constructive and digital

Digital (36, 172, 56)-net over F₄, using

t-expansion [i] based on digital (33, 172, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(172−136, 172, 65)-Net over F₄ — Digital

Digital (36, 172, 65)-net over F₄, using

t-expansion [i] based on digital (33, 172, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(172−136, 172, 148)-Net in Base 4 — Upper bound on s

There is no (36, 172, 149)-net in base 4, because

41 times m-reduction [i] would yield (36, 131, 149)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹³¹, 149, S₄, 95), but
  - the linear programming bound shows that MÂ â‰¥ 33368 039747 905148 197531 005626 408113 677562 887057 731053 159021 618938 897974 262171 069449 240576 / 3975 943153 > 4¹³¹ [i]