MinT - Best Known (252−113, 252, s)-Nets in Base 4

Best Known (252−113, 252, s)-Nets in Base 4

(252−113, 252, 131)-Net over F₄ — Constructive and digital

Digital (139, 252, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 66, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 186, 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]

(252−113, 252, 249)-Net over F₄ — Digital

Digital (139, 252, 249)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(252−113, 252, 3568)-Net in Base 4 — Upper bound on s

There is no (139, 252, 3569)-net in base 4, because

1 times m-reduction [i] would yield (139, 251, 3569)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 200829 833810 594369 934943 280234 481548 942384 001762 749328 036160 297105 679931 589226 215986 271804 532993 652441 380743 952446 669836 976939 947360 885042 272580 494952 > 4²⁵¹ [i]