MinT - Best Known (225−83, 225, s)-Nets in Base 4

Best Known (225−83, 225, s)-Nets in Base 4

(225−83, 225, 147)-Net over F₄ — Constructive and digital

Digital (142, 225, 147)-net over F₄, using

4¹ times duplication [i] based on 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]

(225−83, 225, 152)-Net in Base 4 — Constructive

(142, 225, 152)-net in base 4, using

3 times m-reduction [i] based on (142, 228, 152)-net in base 4, using
- trace code for nets [i] based on (28, 114, 76)-net in base 16, using
  - 1 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]

(225−83, 225, 426)-Net over F₄ — Digital

Digital (142, 225, 426)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(225−83, 225, 10439)-Net in Base 4 — Upper bound on s

There is no (142, 225, 10440)-net in base 4, because

1 times m-reduction [i] would yield (142, 224, 10440)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 726 977197 984596 000509 222275 909381 211924 379994 284594 223227 434885 023795 539853 612921 061249 179451 097125 507663 266694 963395 190990 083928 938120 > 4²²⁴ [i]