MinT - Best Known (129, 129+65, s)-Nets in Base 4

Best Known (129, 129+65, s)-Nets in Base 4

(129, 129+65, 163)-Net over F₄ — Constructive and digital

Digital (129, 194, 163)-net over F₄, using

3 times m-reduction [i] based on digital (129, 197, 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 (80, 148, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 74, 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]

(129, 129+65, 240)-Net in Base 4 — Constructive

(129, 194, 240)-net in base 4, using

2 times m-reduction [i] based on (129, 196, 240)-net in base 4, using
- trace code for nets [i] based on (31, 98, 120)-net in base 16, using
  - 2 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]

(129, 129+65, 525)-Net over F₄ — Digital

Digital (129, 194, 525)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(129, 129+65, 18210)-Net in Base 4 — Upper bound on s

There is no (129, 194, 18211)-net in base 4, because

1 times m-reduction [i] would yield (129, 193, 18211)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 157 882198 881321 667567 342425 197996 947959 735620 337053 150753 261405 812298 968198 772758 415025 896068 262370 024565 325550 667225 > 4¹⁹³ [i]