MinT - Best Known (217−76, 217, s)-Nets in Base 4

Best Known (217−76, 217, s)-Nets in Base 4

(217−76, 217, 163)-Net over F₄ — Constructive and digital

Digital (141, 217, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 53, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (88, 164, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 82, 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]

(217−76, 217, 208)-Net in Base 4 — Constructive

(141, 217, 208)-net in base 4, using

3 times m-reduction [i] based on (141, 220, 208)-net in base 4, using
- trace code for nets [i] based on (31, 110, 104)-net in base 16, using
  - base change [i] based on digital (9, 88, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(217−76, 217, 495)-Net over F₄ — Digital

Digital (141, 217, 495)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(217−76, 217, 13701)-Net in Base 4 — Upper bound on s

There is no (141, 217, 13702)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44395 698691 612090 680149 321504 891624 362887 506144 218182 709325 440792 336820 616613 876276 275440 826445 636870 877643 820781 668738 266516 916384 > 4²¹⁷ [i]