MinT - Best Known (199−73, 199, s)-Nets in Base 4

Best Known (199−73, 199, s)-Nets in Base 4

(199−73, 199, 147)-Net over F₄ — Constructive and digital

Digital (126, 199, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 41, 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 (85, 158, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 79, 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]

(199−73, 199, 152)-Net in Base 4 — Constructive

(126, 199, 152)-net in base 4, using

t-expansion [i] based on (125, 199, 152)-net in base 4, using
- 1 times m-reduction [i] based on (125, 200, 152)-net in base 4, using
  - trace code for nets [i] based on (25, 100, 76)-net in base 16, using
    - base change [i] based on digital (5, 80, 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]

(199−73, 199, 391)-Net over F₄ — Digital

Digital (126, 199, 391)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(199−73, 199, 9719)-Net in Base 4 — Upper bound on s

There is no (126, 199, 9720)-net in base 4, because

1 times m-reduction [i] would yield (126, 198, 9720)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 161860 004834 043070 238713 177394 648982 851966 166002 927755 027833 406086 386887 283431 574411 252255 798809 043965 989220 551099 037795 > 4¹⁹⁸ [i]