MinT - Best Known (215−81, 215, s)-Nets in Base 4

Best Known (215−81, 215, s)-Nets in Base 4

(215−81, 215, 139)-Net over F₄ — Constructive and digital

Digital (134, 215, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 41, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [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]

(215−81, 215, 381)-Net over F₄ — Digital

Digital (134, 215, 381)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(215−81, 215, 8710)-Net in Base 4 — Upper bound on s

There is no (134, 215, 8711)-net in base 4, because

1 times m-reduction [i] would yield (134, 214, 8711)-net in base 4, but
- 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]