MinT - Best Known (243−90, 243, s)-Nets in Base 4

Best Known (243−90, 243, s)-Nets in Base 4

(243−90, 243, 160)-Net over F₄ — Constructive and digital

Digital (153, 243, 160)-net over F₄, using

4 times m-reduction [i] based on digital (153, 247, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 80, 56)-net over F₄, using
    - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 167, 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]

(243−90, 243, 450)-Net over F₄ — Digital

Digital (153, 243, 450)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−90, 243, 10438)-Net in Base 4 — Upper bound on s

There is no (153, 243, 10439)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 200 146705 712348 537286 704260 412100 151654 093274 376906 038760 037801 487035 503855 203236 863591 719207 029150 455138 238012 167340 116043 206697 278517 936422 534640 > 4²⁴³ [i]