MinT - Best Known (252−141, 252, s)-Nets in Base 4

Best Known (252−141, 252, s)-Nets in Base 4

(252−141, 252, 130)-Net over F₄ — Constructive and digital

Digital (111, 252, 130)-net over F₄, using

t-expansion [i] based on digital (105, 252, 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]

(252−141, 252, 165)-Net over F₄ — Digital

Digital (111, 252, 165)-net over F₄, using

t-expansion [i] based on digital (109, 252, 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]

(252−141, 252, 1235)-Net in Base 4 — Upper bound on s

There is no (111, 252, 1236)-net in base 4, because

1 times m-reduction [i] would yield (111, 251, 1236)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 455127 250073 501150 354446 354507 545731 270572 381052 917967 493768 158397 400973 405168 232220 003992 190541 374615 371261 175226 095756 734894 857915 344908 137988 300756 > 4²⁵¹ [i]