MinT - Best Known (227−89, 227, s)-Nets in Base 4

Best Known (227−89, 227, s)-Nets in Base 4

(227−89, 227, 138)-Net over F₄ — Constructive and digital

Digital (138, 227, 138)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 65, 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, 162, 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]

(227−89, 227, 349)-Net over F₄ — Digital

Digital (138, 227, 349)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−89, 227, 7079)-Net in Base 4 — Upper bound on s

There is no (138, 227, 7080)-net in base 4, because

1 times m-reduction [i] would yield (138, 226, 7080)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11678 207400 165079 539687 289813 878243 535598 215537 033249 601564 626944 514857 376093 559146 533462 942721 365309 715958 361679 425838 806752 310350 240272 > 4²²⁶ [i]