MinT - Best Known (199−67, 199, s)-Nets in Base 4

Best Known (199−67, 199, s)-Nets in Base 4

(199−67, 199, 163)-Net over F₄ — Constructive and digital

Digital (132, 199, 163)-net over F₄, using

3 times m-reduction [i] based on digital (132, 202, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 50, 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 (82, 152, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 76, 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]

(199−67, 199, 240)-Net in Base 4 — Constructive

(132, 199, 240)-net in base 4, using

t-expansion [i] based on (131, 199, 240)-net in base 4, using
- 1 times m-reduction [i] based on (131, 200, 240)-net in base 4, using
  - trace code for nets [i] based on (31, 100, 120)-net in base 16, using
    - base change [i] based on digital (11, 80, 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]

(199−67, 199, 528)-Net over F₄ — Digital

Digital (132, 199, 528)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(199−67, 199, 17944)-Net in Base 4 — Upper bound on s

There is no (132, 199, 17945)-net in base 4, because

1 times m-reduction [i] would yield (132, 198, 17945)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 161430 217562 680170 827702 168588 910264 163722 129376 796122 708504 919989 757569 949845 541941 544396 634835 808449 511835 185976 669760 > 4¹⁹⁸ [i]