MinT - Best Known (228−95, 228, s)-Nets in Base 4

Best Known (228−95, 228, s)-Nets in Base 4

(228−95, 228, 134)-Net over F₄ — Constructive and digital

Digital (133, 228, 134)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 60, 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, 168, 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]

(228−95, 228, 288)-Net over F₄ — Digital

Digital (133, 228, 288)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−95, 228, 4913)-Net in Base 4 — Upper bound on s

There is no (133, 228, 4914)-net in base 4, because

1 times m-reduction [i] would yield (133, 227, 4914)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 46665 369857 511537 861735 379471 461823 264965 770359 956233 237402 395273 435909 094680 281019 025633 583868 842522 725487 876757 423998 113323 141965 039400 > 4²²⁷ [i]