MinT - Best Known (226−78, 226, s)-Nets in Base 4

Best Known (226−78, 226, s)-Nets in Base 4

(226−78, 226, 163)-Net over F₄ — Constructive and digital

Digital (148, 226, 163)-net over F₄, using

3 times m-reduction [i] based on digital (148, 229, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 55, 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 (93, 174, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 87, 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]

(226−78, 226, 240)-Net in Base 4 — Constructive

(148, 226, 240)-net in base 4, using

2 times m-reduction [i] based on (148, 228, 240)-net in base 4, using
- trace code for nets [i] based on (34, 114, 120)-net in base 16, using
  - 1 times m-reduction [i] based on (34, 115, 120)-net in base 16, using
    - base change [i] based on digital (11, 92, 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]

(226−78, 226, 541)-Net over F₄ — Digital

Digital (148, 226, 541)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(226−78, 226, 15786)-Net in Base 4 — Upper bound on s

There is no (148, 226, 15787)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11643 922991 919583 967276 674577 963549 666719 868247 347255 850631 623123 947025 186463 441910 703394 459767 056517 705369 936762 784381 931343 764081 414944 > 4²²⁶ [i]