MinT - Best Known (124, 124+71, s)-Nets in Base 4

Best Known (124, 124+71, s)-Nets in Base 4

(124, 124+71, 147)-Net over F₄ — Constructive and digital

Digital (124, 195, 147)-net over F₄, using

4¹ times duplication [i] based on digital (123, 194, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 40, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (83, 154, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 77, 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]

(124, 124+71, 152)-Net in Base 4 — Constructive

(124, 195, 152)-net in base 4, using

3 times m-reduction [i] based on (124, 198, 152)-net in base 4, using
- trace code for nets [i] based on (25, 99, 76)-net in base 16, using
  - 1 times m-reduction [i] based on (25, 100, 76)-net in base 16, using
    - base change [i] based on digital (5, 80, 76)-net over F₃₂, using
      - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(124, 124+71, 395)-Net over F₄ — Digital

Digital (124, 195, 395)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(124, 124+71, 10047)-Net in Base 4 — Upper bound on s

There is no (124, 195, 10048)-net in base 4, because

1 times m-reduction [i] would yield (124, 194, 10048)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 632 492460 273313 891735 425100 986080 430413 378672 251065 026882 793931 127373 196351 383604 212381 048110 161811 158140 764666 969947 > 4¹⁹⁴ [i]