MinT - Best Known (162, 162+83, s)-Nets in Base 4

Best Known (162, 162+83, s)-Nets in Base 4

(162, 162+83, 225)-Net over F₄ — Constructive and digital

Digital (162, 245, 225)-net over F₄, using

base reduction for projective spaces (embedding PG(122,16) in PG(244,4)) for nets [i] based on digital (40, 123, 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]

(162, 162+83, 240)-Net in Base 4 — Constructive

(162, 245, 240)-net in base 4, using

t-expansion [i] based on (161, 245, 240)-net in base 4, using
- 5 times m-reduction [i] based on (161, 250, 240)-net in base 4, using
  - trace code for nets [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]

(162, 162+83, 620)-Net over F₄ — Digital

Digital (162, 245, 620)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(162, 162+83, 20562)-Net in Base 4 — Upper bound on s

There is no (162, 245, 20563)-net in base 4, because

1 times m-reduction [i] would yield (162, 244, 20563)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 800 200576 346054 659504 639001 412144 579437 577135 315704 715587 915856 260289 773996 428693 847103 517020 658348 161738 023182 428097 218889 434035 386617 116084 091400 > 4²⁴⁴ [i]