MinT - Best Known (126, 126+81, s)-Nets in Base 4

Best Known (126, 126+81, s)-Nets in Base 4

(126, 126+81, 134)-Net over F₄ — Constructive and digital

Digital (126, 207, 134)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 53, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 154, 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]

(126, 126+81, 324)-Net over F₄ — Digital

Digital (126, 207, 324)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+81, 6593)-Net in Base 4 — Upper bound on s

There is no (126, 207, 6594)-net in base 4, because

1 times m-reduction [i] would yield (126, 206, 6594)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10599 466050 709153 340682 990042 442641 315922 462965 336425 727489 662090 829725 362039 468039 658587 479112 632626 610726 384812 408654 405656 > 4²⁰⁶ [i]