MinT - Best Known (213−57, 213, s)-Nets in Base 4

Best Known (213−57, 213, s)-Nets in Base 4

(213−57, 213, 531)-Net over F₄ — Constructive and digital

Digital (156, 213, 531)-net over F₄, using

t-expansion [i] based on digital (155, 213, 531)-net over F₄, using
- 9 times m-reduction [i] based on digital (155, 222, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 74, 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]

(213−57, 213, 576)-Net in Base 4 — Constructive

(156, 213, 576)-net in base 4, using

4³ times duplication [i] based on (153, 210, 576)-net in base 4, using
- trace code for nets [i] based on (13, 70, 192)-net in base 64, using
  - base change [i] based on digital (3, 60, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(213−57, 213, 1439)-Net over F₄ — Digital

Digital (156, 213, 1439)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−57, 213, 136227)-Net in Base 4 — Upper bound on s

There is no (156, 213, 136228)-net in base 4, because

1 times m-reduction [i] would yield (156, 212, 136228)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43 326225 859940 837215 414763 811231 102104 356820 011197 135412 761027 753992 877927 906187 847504 570390 087967 846624 182526 890919 483064 096360 > 4²¹² [i]