MinT - Best Known (217−72, 217, s)-Nets in Base 4

Best Known (217−72, 217, s)-Nets in Base 4

(217−72, 217, 195)-Net over F₄ — Constructive and digital

Digital (145, 217, 195)-net over F₄, using

4¹ times duplication [i] based on digital (144, 216, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 72, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(217−72, 217, 240)-Net in Base 4 — Constructive

(145, 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]

(217−72, 217, 601)-Net over F₄ — Digital

Digital (145, 217, 601)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(217−72, 217, 20233)-Net in Base 4 — Upper bound on s

There is no (145, 217, 20234)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44415 842179 680758 958262 507923 613745 651070 428964 438570 893278 951195 381971 865911 231334 502449 191699 751958 089033 526167 142714 901992 581360 > 4²¹⁷ [i]