MinT - Best Known (213−74, 213, s)-Nets in Base 4

Best Known (213−74, 213, s)-Nets in Base 4

(213−74, 213, 163)-Net over F₄ — Constructive and digital

Digital (139, 213, 163)-net over F₄, using

1 times m-reduction [i] based on digital (139, 214, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 52, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (87, 162, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 81, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(213−74, 213, 208)-Net in Base 4 — Constructive

(139, 213, 208)-net in base 4, using

3 times m-reduction [i] based on (139, 216, 208)-net in base 4, using
- trace code for nets [i] based on (31, 108, 104)-net in base 16, using
  - 2 times m-reduction [i] based on (31, 110, 104)-net in base 16, using
    - base change [i] based on digital (9, 88, 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]

(213−74, 213, 501)-Net over F₄ — Digital

Digital (139, 213, 501)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−74, 213, 14248)-Net in Base 4 — Upper bound on s

There is no (139, 213, 14249)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 173 346564 826415 951325 684698 476347 855149 956635 326900 176551 136705 143417 411232 963349 250143 024766 995008 118790 506498 708994 146918 859040 > 4²¹³ [i]