MinT - Best Known (224−83, 224, s)-Nets in Base 4

Best Known (224−83, 224, s)-Nets in Base 4

(224−83, 224, 147)-Net over F₄ — Constructive and digital

Digital (141, 224, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 46, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (95, 178, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 89, 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]

(224−83, 224, 152)-Net in Base 4 — Constructive

(141, 224, 152)-net in base 4, using

2 times m-reduction [i] based on (141, 226, 152)-net in base 4, using
- trace code for nets [i] based on (28, 113, 76)-net in base 16, using
  - 2 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]

(224−83, 224, 418)-Net over F₄ — Digital

Digital (141, 224, 418)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(224−83, 224, 10091)-Net in Base 4 — Upper bound on s

There is no (141, 224, 10092)-net in base 4, because

1 times m-reduction [i] would yield (141, 223, 10092)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 181 901336 080122 091983 498767 930996 861344 674807 852146 310529 261942 198931 329120 285679 215631 793658 808233 385025 349670 519800 971687 630470 084887 > 4²²³ [i]