MinT - Best Known (197−65, 197, s)-Nets in Base 4

Best Known (197−65, 197, s)-Nets in Base 4

(197−65, 197, 195)-Net over F₄ — Constructive and digital

Digital (132, 197, 195)-net over F₄, using

1 times m-reduction [i] based on digital (132, 198, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 66, 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]

(197−65, 197, 240)-Net in Base 4 — Constructive

(132, 197, 240)-net in base 4, using

t-expansion [i] based on (131, 197, 240)-net in base 4, using
- 3 times m-reduction [i] based on (131, 200, 240)-net in base 4, using
  - trace code for nets [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]

(197−65, 197, 564)-Net over F₄ — Digital

Digital (132, 197, 564)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(197−65, 197, 20740)-Net in Base 4 — Upper bound on s

There is no (132, 197, 20741)-net in base 4, because

1 times m-reduction [i] would yield (132, 196, 20741)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10087 228980 123941 106167 169985 514613 514283 546170 584117 366229 857316 591335 308831 048807 009601 528561 196386 953906 092216 531768 > 4¹⁹⁶ [i]