MinT - Best Known (193−121, 193, s)-Nets in Base 4

Best Known (193−121, 193, s)-Nets in Base 4

(193−121, 193, 66)-Net over F₄ — Constructive and digital

Digital (72, 193, 66)-net over F₄, using

t-expansion [i] based on digital (49, 193, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(193−121, 193, 105)-Net over F₄ — Digital

Digital (72, 193, 105)-net over F₄, using

t-expansion [i] based on digital (70, 193, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(193−121, 193, 604)-Net in Base 4 — Upper bound on s

There is no (72, 193, 605)-net in base 4, because

1 times m-reduction [i] would yield (72, 192, 605)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 40 432722 192122 877990 870583 661471 534617 080824 030716 986541 812509 861045 599234 315452 671140 678328 193726 491215 024303 654426 > 4¹⁹² [i]