MinT - Best Known (175−63, 175, s)-Nets in Base 4

Best Known (175−63, 175, s)-Nets in Base 4

(175−63, 175, 147)-Net over F₄ — Constructive and digital

Digital (112, 175, 147)-net over F₄, using

4¹ times duplication [i] based on digital (111, 174, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 36, 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 (75, 138, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 69, 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]

(175−63, 175, 152)-Net in Base 4 — Constructive

(112, 175, 152)-net in base 4, using

3 times m-reduction [i] based on (112, 178, 152)-net in base 4, using
- trace code for nets [i] based on (23, 89, 76)-net in base 16, using
  - 1 times m-reduction [i] based on (23, 90, 76)-net in base 16, using
    - base change [i] based on digital (5, 72, 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]

(175−63, 175, 373)-Net over F₄ — Digital

Digital (112, 175, 373)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(175−63, 175, 9887)-Net in Base 4 — Upper bound on s

There is no (112, 175, 9888)-net in base 4, because

1 times m-reduction [i] would yield (112, 174, 9888)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 573 474847 440232 315863 213363 165701 700455 565770 098899 343291 165227 371784 959524 450274 607353 001687 854419 287895 > 4¹⁷⁴ [i]