MinT - Best Known (193−177, 193, s)-Nets in Base 4

Best Known (193−177, 193, s)-Nets in Base 4

(193−177, 193, 33)-Net over F₄ — Constructive and digital

Digital (16, 193, 33)-net over F₄, using

t-expansion [i] based on digital (15, 193, 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]

(193−177, 193, 36)-Net over F₄ — Digital

Digital (16, 193, 36)-net over F₄, using

net from sequence [i] based on digital (16, 35)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 16 and N(F) ≥ 36, using
  - Drinfeld modules of rank 1 [i]

(193−177, 193, 59)-Net in Base 4 — Upper bound on s

There is no (16, 193, 60)-net in base 4, because

17 times m-reduction [i] would yield (16, 176, 60)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁷⁶, 60, S₄, 3, 160), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1 614623 025657 010344 174017 910292 669206 320781 752374 469911 111262 542928 393704 766293 304881 293061 389528 483334 455296 / 161 > 4¹⁷⁶ [i]