MinT - Best Known (214−80, 214, s)-Nets in Base 4

Best Known (214−80, 214, s)-Nets in Base 4

(214−80, 214, 140)-Net over F₄ — Constructive and digital

Digital (134, 214, 140)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 42, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (92, 172, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 86, 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]

(214−80, 214, 152)-Net in Base 4 — Constructive

(134, 214, 152)-net in base 4, using

trace code for nets [i] based on (27, 107, 76)-net in base 16, using
- 3 times m-reduction [i] based on (27, 110, 76)-net in base 16, using
  - base change [i] based on digital (5, 88, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(214−80, 214, 390)-Net over F₄ — Digital

Digital (134, 214, 390)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(214−80, 214, 8710)-Net in Base 4 — Upper bound on s

There is no (134, 214, 8711)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 693 723232 305714 171353 353910 332085 456042 695047 194289 142430 020728 501087 456782 701763 370700 628628 032676 077832 634761 298511 579205 332800 > 4²¹⁴ [i]