MinT - Best Known (144, 221, s)-Nets in Base 4

Best Known (144, 221, s)-Nets in Base 4

(144, 221, 163)-Net over F₄ — Constructive and digital

Digital (144, 221, 163)-net over F₄, using

1 times m-reduction [i] based on digital (144, 222, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 54, 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 (90, 168, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 84, 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]

(144, 221, 240)-Net in Base 4 — Constructive

(144, 221, 240)-net in base 4, using

4¹ times duplication [i] based on (143, 220, 240)-net in base 4, using
- trace code for nets [i] based on (33, 110, 120)-net in base 16, using
  - base change [i] based on digital (11, 88, 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]

(144, 221, 513)-Net over F₄ — Digital

Digital (144, 221, 513)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(144, 221, 15290)-Net in Base 4 — Upper bound on s

There is no (144, 221, 15291)-net in base 4, because

1 times m-reduction [i] would yield (144, 220, 15291)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 845954 591854 678901 003539 314091 779785 570919 034194 844304 821122 650239 292265 553571 102906 474829 275887 847642 696221 318388 201715 329576 228090 > 4²²⁰ [i]