MinT - Best Known (172−163, 172, s)-Nets in Base 4

Best Known (172−163, 172, s)-Nets in Base 4

(172−163, 172, 22)-Net over F₄ — Constructive and digital

Digital (9, 172, 22)-net over F₄, using

net from sequence [i] based on digital (9, 21)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
  - an explicitly constructive algebraic function field [i]

(172−163, 172, 26)-Net over F₄ — Digital

Digital (9, 172, 26)-net over F₄, using

net from sequence [i] based on digital (9, 25)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 26, using
  - Drinfeld modules of rank 1 [i]

(172−163, 172, 37)-Net in Base 4 — Upper bound on s

There is no (9, 172, 38)-net in base 4, because

62 times m-reduction [i] would yield (9, 110, 38)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹¹⁰, 38, S₄, 3, 101), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 33 699933 333938 299743 333768 858774 538342 046430 528175 715601 379512 811520 / 17 > 4¹¹⁰ [i]