MinT - Best Known (249−85, 249, s)-Nets in Base 4

Best Known (249−85, 249, s)-Nets in Base 4

(249−85, 249, 225)-Net over F₄ — Constructive and digital

Digital (164, 249, 225)-net over F₄, using

base reduction for projective spaces (embedding PG(124,16) in PG(248,4)) for nets [i] based on digital (40, 125, 225)-net over F₁₆, using
- net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
    - a function field by GarcÃa and Quoos [i]

(249−85, 249, 240)-Net in Base 4 — Constructive

(164, 249, 240)-net in base 4, using

5 times m-reduction [i] based on (164, 254, 240)-net in base 4, using
- trace code for nets [i] based on (37, 127, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (37, 130, 120)-net in base 16, using
    - base change [i] based on digital (11, 104, 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]

(249−85, 249, 613)-Net over F₄ — Digital

Digital (164, 249, 613)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−85, 249, 19721)-Net in Base 4 — Upper bound on s

There is no (164, 249, 19722)-net in base 4, because

1 times m-reduction [i] would yield (164, 248, 19722)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 204745 119051 960886 873894 872123 534203 218250 728608 564811 017245 571073 307076 231878 554913 385868 973045 131381 786870 465822 500642 560722 105414 586404 535437 449280 > 4²⁴⁸ [i]