MinT - Best Known (132, 132+69, s)-Nets in Base 4

Best Known (132, 132+69, s)-Nets in Base 4

(132, 132+69, 163)-Net over F₄ — Constructive and digital

Digital (132, 201, 163)-net over F₄, using

1 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]

(132, 132+69, 240)-Net in Base 4 — Constructive

(132, 201, 240)-net in base 4, using

4¹ times duplication [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]

(132, 132+69, 497)-Net over F₄ — Digital

Digital (132, 201, 497)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(132, 132+69, 15671)-Net in Base 4 — Upper bound on s

There is no (132, 201, 15672)-net in base 4, because

1 times m-reduction [i] would yield (132, 200, 15672)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 584471 116958 124624 816779 793218 368872 711398 225450 921569 038904 035939 420220 881935 229822 334310 702423 833000 559137 283373 558083 > 4²⁰⁰ [i]