MinT - Best Known (219−78, 219, s)-Nets in Base 4

Best Known (219−78, 219, s)-Nets in Base 4

(219−78, 219, 158)-Net over F₄ — Constructive and digital

Digital (141, 219, 158)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 51, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (90, 168, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 84, 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]

(219−78, 219, 208)-Net in Base 4 — Constructive

(141, 219, 208)-net in base 4, using

1 times m-reduction [i] based on (141, 220, 208)-net in base 4, using
- trace code for nets [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]

(219−78, 219, 472)-Net over F₄ — Digital

Digital (141, 219, 472)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−78, 219, 12301)-Net in Base 4 — Upper bound on s

There is no (141, 219, 12302)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 709829 458677 037291 651843 575931 090244 303445 960357 771269 993171 578832 324632 287740 991259 063996 606250 758665 110636 243561 978120 789742 568960 > 4²¹⁹ [i]