MinT - Best Known (217−73, 217, s)-Nets in Base 4

Best Known (217−73, 217, s)-Nets in Base 4

(217−73, 217, 164)-Net over F₄ — Constructive and digital

Digital (144, 217, 164)-net over F₄, using

1 times m-reduction [i] based on digital (144, 218, 164)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 58, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (86, 160, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 80, 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]

(217−73, 217, 240)-Net in Base 4 — Constructive

(144, 217, 240)-net in base 4, using

t-expansion [i] based on (143, 217, 240)-net in base 4, using
- 3 times m-reduction [i] based on (143, 220, 240)-net in base 4, using
  - trace code for nets [i] based on (33, 110, 120)-net in base 16, using
    - base change [i] based on digital (11, 88, 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]

(217−73, 217, 571)-Net over F₄ — Digital

Digital (144, 217, 571)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(217−73, 217, 19467)-Net in Base 4 — Upper bound on s

There is no (144, 217, 19468)-net in base 4, because

1 times m-reduction [i] would yield (144, 216, 19468)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11093 222954 145064 382870 412281 339707 237432 556964 882486 388233 716636 742532 759442 341853 370474 858496 063421 321392 819141 889709 890356 056640 > 4²¹⁶ [i]