MinT - Best Known (232−92, 232, s)-Nets in Base 4

Best Known (232−92, 232, s)-Nets in Base 4

(232−92, 232, 138)-Net over F₄ — Constructive and digital

Digital (140, 232, 138)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 67, 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, 165, 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]

(232−92, 232, 343)-Net over F₄ — Digital

Digital (140, 232, 343)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(232−92, 232, 6487)-Net in Base 4 — Upper bound on s

There is no (140, 232, 6488)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47 863745 352327 693881 239837 980645 394316 898117 670569 541731 654921 447123 301531 296619 016269 947535 036518 392235 758493 131312 024817 664460 002963 477540 > 4²³² [i]