MinT - Best Known (252−171, 252, s)-Nets in Base 4

Best Known (252−171, 252, s)-Nets in Base 4

(252−171, 252, 104)-Net over F₄ — Constructive and digital

Digital (81, 252, 104)-net over F₄, using

t-expansion [i] based on digital (73, 252, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(252−171, 252, 129)-Net over F₄ — Digital

Digital (81, 252, 129)-net over F₄, using

net from sequence [i] based on digital (81, 128)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
  - Drinfeld modules of rank 1 [i]

(252−171, 252, 580)-Net in Base 4 — Upper bound on s

There is no (81, 252, 581)-net in base 4, because

1 times m-reduction [i] would yield (81, 251, 581)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 256599 627069 770973 685409 968585 308462 663796 173251 533777 744647 944971 780822 206826 792513 537109 613247 187642 718124 501026 636381 389501 012998 232128 676689 875840 > 4²⁵¹ [i]