MinT - Best Known (253−109, 253, s)-Nets in Base 4

Best Known (253−109, 253, s)-Nets in Base 4

(253−109, 253, 137)-Net over F₄ — Constructive and digital

Digital (144, 253, 137)-net over F₄, using

3 times m-reduction [i] based on digital (144, 256, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 71, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 185, 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]

(253−109, 253, 283)-Net over F₄ — Digital

Digital (144, 253, 283)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−109, 253, 4464)-Net in Base 4 — Upper bound on s

There is no (144, 253, 4465)-net in base 4, because

1 times m-reduction [i] would yield (144, 252, 4465)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 880302 800087 079018 859010 716699 636308 174417 787645 972874 075490 834983 808659 171482 385339 944545 461221 113143 949509 811029 473459 775843 702872 277803 702523 301600 > 4²⁵² [i]