MinT - Best Known (199−69, 199, s)-Nets in Base 4

Best Known (199−69, 199, s)-Nets in Base 4

(199−69, 199, 163)-Net over F₄ — Constructive and digital

Digital (130, 199, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 49, 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 (81, 150, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 75, 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]

(199−69, 199, 208)-Net in Base 4 — Constructive

(130, 199, 208)-net in base 4, using

t-expansion [i] based on (129, 199, 208)-net in base 4, using
- 1 times m-reduction [i] based on (129, 200, 208)-net in base 4, using
  - trace code for nets [i] based on (29, 100, 104)-net in base 16, using
    - base change [i] based on digital (9, 80, 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]

(199−69, 199, 475)-Net over F₄ — Digital

Digital (130, 199, 475)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(199−69, 199, 14442)-Net in Base 4 — Upper bound on s

There is no (130, 199, 14443)-net in base 4, because

1 times m-reduction [i] would yield (130, 198, 14443)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 161702 791528 654962 263813 168395 336944 926554 317265 172564 999011 228008 557148 333112 750516 415838 953581 610388 383801 266480 602175 > 4¹⁹⁸ [i]