MinT - Best Known (53, 53+112, s)-Nets in Base 3

Best Known (53, 53+112, s)-Nets in Base 3

(53, 53+112, 48)-Net over F₃ — Constructive and digital

Digital (53, 165, 48)-net over F₃, using

t-expansion [i] based on digital (45, 165, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(53, 53+112, 64)-Net over F₃ — Digital

Digital (53, 165, 64)-net over F₃, using

t-expansion [i] based on digital (49, 165, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(53, 53+112, 166)-Net over F₃ — Upper bound on s (digital)

There is no digital (53, 165, 167)-net over F₃, because

4 times m-reduction [i] would yield digital (53, 161, 167)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁶¹, 167, F₃, 108) (dual of [167, 6, 109]-code), but
  - residual code [i] would yield linear OA(3⁵³, 58, F₃, 36) (dual of [58, 5, 37]-code), but
    - residual code [i] would yield linear OA(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code), but
      - â€œHNâ€ bound on codes from Brouwerâ€™s database [i]

(53, 53+112, 225)-Net in Base 3 — Upper bound on s

There is no (53, 165, 226)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 494349 488109 764898 218149 423513 917554 320986 858446 499177 244722 503079 760871 605809 > 3¹⁶⁵ [i]