MinT - Best Known (136, 136+103, s)-Nets in Base 4

Best Known (136, 136+103, s)-Nets in Base 4

(136, 136+103, 132)-Net over F₄ — Constructive and digital

Digital (136, 239, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 63, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 176, 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]

(136, 136+103, 269)-Net over F₄ — Digital

Digital (136, 239, 269)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+103, 4227)-Net in Base 4 — Upper bound on s

There is no (136, 239, 4228)-net in base 4, because

1 times m-reduction [i] would yield (136, 238, 4228)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 196582 517459 598988 196082 939035 861456 236452 738313 587242 766810 293952 202081 738710 140335 077838 538349 620612 713394 504956 259503 631639 641627 243132 493560 > 4²³⁸ [i]