MinT - Best Known (126, 126+62, s)-Nets in Base 4

Best Known (126, 126+62, s)-Nets in Base 4

(126, 126+62, 195)-Net over F₄ — Constructive and digital

Digital (126, 188, 195)-net over F₄, using

1 times m-reduction [i] based on digital (126, 189, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 63, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(126, 126+62, 240)-Net in Base 4 — Constructive

(126, 188, 240)-net in base 4, using

t-expansion [i] based on (125, 188, 240)-net in base 4, using
- 2 times m-reduction [i] based on (125, 190, 240)-net in base 4, using
  - trace code for nets [i] based on (30, 95, 120)-net in base 16, using
    - base change [i] based on digital (11, 76, 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]

(126, 126+62, 542)-Net over F₄ — Digital

Digital (126, 188, 542)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+62, 18514)-Net in Base 4 — Upper bound on s

There is no (126, 188, 18515)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 153946 971952 323020 653788 711369 595750 411492 957511 263386 530653 272469 608575 617312 322251 835548 519228 992830 704173 061120 > 4¹⁸⁸ [i]