MinT - Best Known (157−52, 157, s)-Nets in Base 4

Best Known (157−52, 157, s)-Nets in Base 4

(157−52, 157, 195)-Net over F₄ — Constructive and digital

Digital (105, 157, 195)-net over F₄, using

4¹ times duplication [i] based on digital (104, 156, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 52, 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]

(157−52, 157, 208)-Net in Base 4 — Constructive

(105, 157, 208)-net in base 4, using

3 times m-reduction [i] based on (105, 160, 208)-net in base 4, using
- trace code for nets [i] based on (25, 80, 104)-net in base 16, using
  - base change [i] based on digital (9, 64, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(157−52, 157, 458)-Net over F₄ — Digital

Digital (105, 157, 458)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−52, 157, 15173)-Net in Base 4 — Upper bound on s

There is no (105, 157, 15174)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 33409 296408 848710 391606 754853 328985 716966 085923 572815 384498 549566 167052 985076 011973 864507 298896 > 4¹⁵⁷ [i]