MinT - Best Known (144, 144+113, s)-Nets in Base 4

Best Known (144, 144+113, s)-Nets in Base 4

(144, 144+113, 137)-Net over F₄ — Constructive and digital

Digital (144, 257, 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, 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]

(144, 144+113, 269)-Net over F₄ — Digital

Digital (144, 257, 269)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(144, 144+113, 4044)-Net in Base 4 — Upper bound on s

There is no (144, 257, 4045)-net in base 4, because

1 times m-reduction [i] would yield (144, 256, 4045)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13464 367913 155475 822939 309908 143813 132593 791689 673316 930461 992892 183506 640380 978296 218847 645569 665546 029930 940158 830964 815557 524845 443388 887995 684284 784980 > 4²⁵⁶ [i]