MinT - Best Known (224−84, 224, s)-Nets in Base 4

Best Known (224−84, 224, s)-Nets in Base 4

(224−84, 224, 140)-Net over F₄ — Constructive and digital

Digital (140, 224, 140)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 44, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (96, 180, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 90, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(224−84, 224, 152)-Net in Base 4 — Constructive

(140, 224, 152)-net in base 4, using

trace code for nets [i] based on (28, 112, 76)-net in base 16, using
- 3 times m-reduction [i] based on (28, 115, 76)-net in base 16, using
  - base change [i] based on digital (5, 92, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(224−84, 224, 401)-Net over F₄ — Digital

Digital (140, 224, 401)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(224−84, 224, 8912)-Net in Base 4 — Upper bound on s

There is no (140, 224, 8913)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 729 092231 515062 635749 226771 822359 526215 315659 915025 259643 836575 939185 671215 916205 197414 705337 657100 429722 528130 123813 324130 824585 825360 > 4²²⁴ [i]