MinT - Best Known (186−63, 186, s)-Nets in Base 4

Best Known (186−63, 186, s)-Nets in Base 4

(186−63, 186, 163)-Net over F₄ — Constructive and digital

Digital (123, 186, 163)-net over F₄, using

1 times m-reduction [i] based on digital (123, 187, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 47, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (76, 140, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 70, 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]

(186−63, 186, 240)-Net in Base 4 — Constructive

(123, 186, 240)-net in base 4, using

trace code for nets [i] based on (30, 93, 120)-net in base 16, using
- 2 times m-reduction [i] based on (30, 95, 120)-net in base 16, using
  - base change [i] based on digital (11, 76, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(186−63, 186, 487)-Net over F₄ — Digital

Digital (123, 186, 487)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(186−63, 186, 16187)-Net in Base 4 — Upper bound on s

There is no (123, 186, 16188)-net in base 4, because

1 times m-reduction [i] would yield (123, 185, 16188)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2408 608212 301237 233125 184843 291303 832903 728270 691425 949156 692846 436728 881545 087604 666236 829464 771051 902524 896600 > 4¹⁸⁵ [i]