MinT - Best Known (13, 165, s)-Nets in Base 4

Best Known (13, 165, s)-Nets in Base 4

(13, 165, 30)-Net over F₄ — Constructive and digital

Digital (13, 165, 30)-net over F₄, using

net from sequence [i] based on digital (13, 29)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
  - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(13, 165, 33)-Net over F₄ — Digital

Digital (13, 165, 33)-net over F₄, using

net from sequence [i] based on digital (13, 32)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 33, using
  - Drinfeld modules of rank 1 [i]

(13, 165, 50)-Net in Base 4 — Upper bound on s

There is no (13, 165, 51)-net in base 4, because

16 times m-reduction [i] would yield (13, 149, 51)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁴⁹, 51, S₄, 3, 136), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 95 740690 887720 846054 616947 355240 773569 419014 502299 003779 898601 119655 921088 876825 190619 676672 / 137 > 4¹⁴⁹ [i]