MinT - Best Known (172−139, 172, s)-Nets in Base 4

Best Known (172−139, 172, s)-Nets in Base 4

(172−139, 172, 56)-Net over F₄ — Constructive and digital

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]

(172−139, 172, 65)-Net over F₄ — Digital

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]

(172−139, 172, 137)-Net in Base 4 — Upper bound on s

There is no (33, 172, 138)-net in base 4, because

49 times m-reduction [i] would yield (33, 123, 138)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹²³, 138, S₄, 90), but
  - the linear programming bound shows that MÂ â‰¥ 7351 408161 498730 615051 674696 231577 893788 404786 452188 129737 083093 494388 774580 256768 / 64 507625 > 4¹²³ [i]