MinT - Best Known (172−62, 172, s)-Nets in Base 4

Best Known (172−62, 172, s)-Nets in Base 4

(172−62, 172, 147)-Net over F₄ — Constructive and digital

Digital (110, 172, 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 (74, 136, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 68, 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]

(172−62, 172, 152)-Net in Base 4 — Constructive

(110, 172, 152)-net in base 4, using

2 times m-reduction [i] based on (110, 174, 152)-net in base 4, using
- trace code for nets [i] based on (23, 87, 76)-net in base 16, using
  - 3 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]

(172−62, 172, 366)-Net over F₄ — Digital

Digital (110, 172, 366)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−62, 172, 9039)-Net in Base 4 — Upper bound on s

There is no (110, 172, 9040)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 35 858646 147175 180700 823466 210547 261329 572843 868274 800747 718179 668060 155080 052523 232551 408003 506587 294780 > 4¹⁷² [i]