MinT - Best Known (221−75, 221, s)-Nets in Base 4

Best Known (221−75, 221, s)-Nets in Base 4

(221−75, 221, 164)-Net over F₄ — Constructive and digital

Digital (146, 221, 164)-net over F₄, using

4¹ times duplication [i] based on digital (145, 220, 164)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 58, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (87, 162, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 81, 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]

(221−75, 221, 240)-Net in Base 4 — Constructive

(146, 221, 240)-net in base 4, using

3 times m-reduction [i] based on (146, 224, 240)-net in base 4, using
- trace code for nets [i] based on (34, 112, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (34, 115, 120)-net in base 16, using
    - base change [i] based on digital (11, 92, 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]

(221−75, 221, 562)-Net over F₄ — Digital

Digital (146, 221, 562)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(221−75, 221, 18530)-Net in Base 4 — Upper bound on s

There is no (146, 221, 18531)-net in base 4, because

1 times m-reduction [i] would yield (146, 220, 18531)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 839699 562222 293192 029242 394292 830043 064484 343784 321243 032522 301323 336104 841070 581421 881310 577975 521011 608234 432726 752495 536276 477600 > 4²²⁰ [i]