MinT - Best Known (193−65, 193, s)-Nets in Base 4

Best Known (193−65, 193, s)-Nets in Base 4

(193−65, 193, 163)-Net over F₄ — Constructive and digital

Digital (128, 193, 163)-net over F₄, using

t-expansion [i] based on digital (127, 193, 163)-net over F₄, using
- 1 times m-reduction [i] based on digital (127, 194, 163)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 48, 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 (79, 146, 130)-net over F₄, using
      - trace code for nets [i] based on digital (6, 73, 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]

(193−65, 193, 240)-Net in Base 4 — Constructive

(128, 193, 240)-net in base 4, using

1 times m-reduction [i] based on (128, 194, 240)-net in base 4, using
- trace code for nets [i] based on (31, 97, 120)-net in base 16, using
  - 3 times m-reduction [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]

(193−65, 193, 513)-Net over F₄ — Digital

Digital (128, 193, 513)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(193−65, 193, 17436)-Net in Base 4 — Upper bound on s

There is no (128, 193, 17437)-net in base 4, because

1 times m-reduction [i] would yield (128, 192, 17437)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 39 412355 323777 451453 330754 253919 783126 686031 418815 408742 230690 089187 989439 464524 031573 140421 787185 511157 011357 140000 > 4¹⁹² [i]