MinT - Best Known (153−51, 153, s)-Nets in Base 4

Best Known (153−51, 153, s)-Nets in Base 4

(153−51, 153, 195)-Net over F₄ — Constructive and digital

Digital (102, 153, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 51, 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]

(153−51, 153, 208)-Net in Base 4 — Constructive

(102, 153, 208)-net in base 4, using

1 times m-reduction [i] based on (102, 154, 208)-net in base 4, using
- trace code for nets [i] based on (25, 77, 104)-net in base 16, using
  - 3 times m-reduction [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]

(153−51, 153, 438)-Net over F₄ — Digital

Digital (102, 153, 438)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(153−51, 153, 15504)-Net in Base 4 — Upper bound on s

There is no (102, 153, 15505)-net in base 4, because

1 times m-reduction [i] would yield (102, 152, 15505)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 32 613833 623881 604576 314004 363073 342819 202140 057182 514655 650912 619754 627387 888223 552591 368384 > 4¹⁵² [i]