MinT - Best Known (257−66, 257, s)-Nets in Base 4

Best Known (257−66, 257, s)-Nets in Base 4

(257−66, 257, 548)-Net over F₄ — Constructive and digital

Digital (191, 257, 548)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 38, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (153, 219, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 73, 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]

(257−66, 257, 648)-Net in Base 4 — Constructive

(191, 257, 648)-net in base 4, using

t-expansion [i] based on (190, 257, 648)-net in base 4, using
- 1 times m-reduction [i] based on (190, 258, 648)-net in base 4, using
  - trace code for nets [i] based on (18, 86, 216)-net in base 64, using
    - 5 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
      - base change [i] based on digital (5, 78, 216)-net over F₁₂₈, using
        net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(257−66, 257, 2037)-Net over F₄ — Digital

Digital (191, 257, 2037)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−66, 257, 214263)-Net in Base 4 — Upper bound on s

There is no (191, 257, 214264)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53634 640145 246380 349712 635220 131276 858348 509982 367034 964298 538211 850848 495801 682562 771573 157833 345779 601735 565553 113770 473583 239945 233338 160265 345494 912605 > 4²⁵⁷ [i]