MinT - Best Known (137, 137+87, s)-Nets in Base 4

Best Known (137, 137+87, s)-Nets in Base 4

(137, 137+87, 138)-Net over F₄ — Constructive and digital

Digital (137, 224, 138)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 64, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 160, 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]

(137, 137+87, 356)-Net over F₄ — Digital

Digital (137, 224, 356)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+87, 7423)-Net in Base 4 — Upper bound on s

There is no (137, 224, 7424)-net in base 4, because

1 times m-reduction [i] would yield (137, 223, 7424)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 182 392839 932396 122705 066070 578318 101652 756845 110752 234162 200610 869886 210105 371509 935350 078525 749436 839184 485519 857421 942716 111671 691369 > 4²²³ [i]