MinT - Best Known (219−109, 219, s)-Nets in Base 4

Best Known (219−109, 219, s)-Nets in Base 4

(219−109, 219, 130)-Net over F₄ — Constructive and digital

Digital (110, 219, 130)-net over F₄, using

t-expansion [i] based on digital (105, 219, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(219−109, 219, 165)-Net over F₄ — Digital

Digital (110, 219, 165)-net over F₄, using

t-expansion [i] based on digital (109, 219, 165)-net over F₄, using
- net from sequence [i] based on digital (109, 164)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
    - Drinfeld modules of rank 1 [i]

(219−109, 219, 1839)-Net in Base 4 — Upper bound on s

There is no (110, 219, 1840)-net in base 4, because

1 times m-reduction [i] would yield (110, 218, 1840)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 180667 813440 530082 699718 593216 931313 931836 834384 092623 837561 433573 060427 993827 629596 485920 564240 831175 994092 572196 323053 738714 696080 > 4²¹⁸ [i]