MinT - Best Known (172, 172+60, s)-Nets in Base 4

Best Known (172, 172+60, s)-Nets in Base 4

(172, 172+60, 540)-Net over F₄ — Constructive and digital

Digital (172, 232, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 31, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (141, 201, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 67, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(172, 172+60, 648)-Net in Base 4 — Constructive

(172, 232, 648)-net in base 4, using

4¹ times duplication [i] based on (171, 231, 648)-net in base 4, using
- t-expansion [i] based on (170, 231, 648)-net in base 4, using
  - trace code for nets [i] based on (16, 77, 216)-net in base 64, using
    - base change [i] based on digital (5, 66, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(172, 172+60, 1802)-Net over F₄ — Digital

Digital (172, 232, 1802)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(172, 172+60, 181777)-Net in Base 4 — Upper bound on s

There is no (172, 232, 181778)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47 640137 406643 448556 252062 866103 279313 726212 314624 573450 187336 871201 081521 251475 259994 935826 172591 936467 269032 573923 239917 071005 485411 477248 > 4²³² [i]