MinT - Best Known (114, 114+57, s)-Nets in Base 4

Best Known (114, 114+57, s)-Nets in Base 4

(114, 114+57, 195)-Net over F₄ — Constructive and digital

Digital (114, 171, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 57, 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]

(114, 114+57, 240)-Net in Base 4 — Constructive

(114, 171, 240)-net in base 4, using

4¹ times duplication [i] based on (113, 170, 240)-net in base 4, using
- trace code for nets [i] based on (28, 85, 120)-net in base 16, using
  - base change [i] based on digital (11, 68, 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]

(114, 114+57, 481)-Net over F₄ — Digital

Digital (114, 171, 481)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(114, 114+57, 17008)-Net in Base 4 — Upper bound on s

There is no (114, 171, 17009)-net in base 4, because

1 times m-reduction [i] would yield (114, 170, 17009)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 241635 867647 338196 792458 285308 253963 321338 618364 511296 390058 302095 793921 876881 881056 701193 227837 286808 > 4¹⁷⁰ [i]