MinT - Best Known (165, 165+58, s)-Nets in Base 4

Best Known (165, 165+58, s)-Nets in Base 4

(165, 165+58, 531)-Net over F₄ — Constructive and digital

Digital (165, 223, 531)-net over F₄, using

14 times m-reduction [i] based on digital (165, 237, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 79, 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]

(165, 165+58, 648)-Net in Base 4 — Constructive

(165, 223, 648)-net in base 4, using

4¹ times duplication [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]

(165, 165+58, 1697)-Net over F₄ — Digital

Digital (165, 223, 1697)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+58, 165795)-Net in Base 4 — Upper bound on s

There is no (165, 223, 165796)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 181 728299 909733 208739 537364 908417 117253 789984 842012 076910 785415 258966 353005 889434 159596 526384 132249 424494 071354 988160 737141 325346 854952 > 4²²³ [i]