MinT - Best Known (228−88, 228, s)-Nets in Base 4

Best Known (228−88, 228, s)-Nets in Base 4

(228−88, 228, 138)-Net over F₄ — Constructive and digital

Digital (140, 228, 138)-net over F₄, using

4 times m-reduction [i] based on digital (140, 232, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 67, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 165, 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]

(228−88, 228, 369)-Net over F₄ — Digital

Digital (140, 228, 369)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−88, 228, 7542)-Net in Base 4 — Upper bound on s

There is no (140, 228, 7543)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 187031 056456 911719 752997 128651 665685 040316 592230 257630 120164 599250 947361 456695 354588 778173 094798 257572 601127 546056 018656 352370 031416 457254 > 4²²⁸ [i]