MinT - Best Known (214−109, 214, s)-Nets in Base 4

Best Known (214−109, 214, s)-Nets in Base 4

(214−109, 214, 130)-Net over F₄ — Constructive and digital

Digital (105, 214, 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]

(214−109, 214, 144)-Net over F₄ — Digital

Digital (105, 214, 144)-net over F₄, using

t-expansion [i] based on digital (91, 214, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(214−109, 214, 1612)-Net in Base 4 — Upper bound on s

There is no (105, 214, 1613)-net in base 4, because

1 times m-reduction [i] would yield (105, 213, 1613)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 175 931861 204312 221183 764376 444491 825187 423382 101337 993698 328606 295707 036423 008550 875925 491667 319001 707744 969032 127942 736686 521296 > 4²¹³ [i]