MinT - Best Known (214−78, 214, s)-Nets in Base 4

Best Known (214−78, 214, s)-Nets in Base 4

(214−78, 214, 151)-Net over F₄ — Constructive and digital

Digital (136, 214, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 46, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (90, 168, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 84, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(214−78, 214, 196)-Net in Base 4 — Constructive

(136, 214, 196)-net in base 4, using

trace code for nets [i] based on (29, 107, 98)-net in base 16, using
- 3 times m-reduction [i] based on (29, 110, 98)-net in base 16, using
  - base change [i] based on digital (7, 88, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(214−78, 214, 426)-Net over F₄ — Digital

Digital (136, 214, 426)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(214−78, 214, 10293)-Net in Base 4 — Upper bound on s

There is no (136, 214, 10294)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 694 119577 451891 475299 310247 187152 268681 642765 199170 432978 653407 051750 992043 186046 273645 226952 080162 610979 622297 356897 233297 329800 > 4²¹⁴ [i]