MinT - Best Known (241−81, 241, s)-Nets in Base 4

Best Known (241−81, 241, s)-Nets in Base 4

(241−81, 241, 225)-Net over F₄ — Constructive and digital

Digital (160, 241, 225)-net over F₄, using

base reduction for projective spaces (embedding PG(120,16) in PG(240,4)) for nets [i] based on digital (40, 121, 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]

(241−81, 241, 240)-Net in Base 4 — Constructive

(160, 241, 240)-net in base 4, using

7 times m-reduction [i] based on (160, 248, 240)-net in base 4, using
- trace code for nets [i] based on (36, 124, 120)-net in base 16, using
  - 1 times m-reduction [i] based on (36, 125, 120)-net in base 16, using
    - base change [i] based on digital (11, 100, 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]

(241−81, 241, 629)-Net over F₄ — Digital

Digital (160, 241, 629)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(241−81, 241, 21496)-Net in Base 4 — Upper bound on s

There is no (160, 241, 21497)-net in base 4, because

1 times m-reduction [i] would yield (160, 240, 21497)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 124757 791042 202370 065865 756796 866732 818584 024161 026165 372044 599147 194412 048578 997898 142233 830935 407993 067164 336707 870828 606419 981710 139969 046571 > 4²⁴⁰ [i]