MinT - Best Known (164, 164+57, s)-Nets in Base 4

Best Known (164, 164+57, s)-Nets in Base 4

(164, 164+57, 540)-Net over F₄ — Constructive and digital

Digital (164, 221, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 29, 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 (135, 192, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 64, 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]

(164, 164+57, 648)-Net in Base 4 — Constructive

(164, 221, 648)-net in base 4, using

1 times m-reduction [i] based on (164, 222, 648)-net in base 4, using
- trace code for nets [i] based on (16, 74, 216)-net in base 64, using
  - 3 times m-reduction [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]

(164, 164+57, 1748)-Net over F₄ — Digital

Digital (164, 221, 1748)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(164, 164+57, 202444)-Net in Base 4 — Upper bound on s

There is no (164, 221, 202445)-net in base 4, because

1 times m-reduction [i] would yield (164, 220, 202445)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 839378 181914 452275 617341 308239 645588 265424 621461 351131 227501 783353 885143 117308 103433 356727 485447 437940 379556 390641 387802 886398 031420 > 4²²⁰ [i]