MinT - Best Known (213−71, 213, s)-Nets in Base 4

Best Known (213−71, 213, s)-Nets in Base 4

(213−71, 213, 195)-Net over F₄ — Constructive and digital

Digital (142, 213, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 71, 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]

(213−71, 213, 240)-Net in Base 4 — Constructive

(142, 213, 240)-net in base 4, using

5 times m-reduction [i] based on (142, 218, 240)-net in base 4, using
- trace code for nets [i] based on (33, 109, 120)-net in base 16, using
  - 1 times m-reduction [i] based on (33, 110, 120)-net in base 16, using
    - base change [i] based on digital (11, 88, 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]

(213−71, 213, 581)-Net over F₄ — Digital

Digital (142, 213, 581)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−71, 213, 20525)-Net in Base 4 — Upper bound on s

There is no (142, 213, 20526)-net in base 4, because

1 times m-reduction [i] would yield (142, 212, 20526)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43 337640 768493 454659 177841 392572 188624 661802 550800 261542 197944 233752 091074 379415 596974 153590 271615 159970 915551 823709 612382 949802 > 4²¹² [i]