MinT - Best Known (162−109, 162, s)-Nets in Base 3

Best Known (162−109, 162, s)-Nets in Base 3

(162−109, 162, 48)-Net over F₃ — Constructive and digital

Digital (53, 162, 48)-net over F₃, using

t-expansion [i] based on digital (45, 162, 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]

(162−109, 162, 64)-Net over F₃ — Digital

Digital (53, 162, 64)-net over F₃, using

t-expansion [i] based on digital (49, 162, 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]

(162−109, 162, 166)-Net over F₃ — Upper bound on s (digital)

There is no digital (53, 162, 167)-net over F₃, because

1 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]

(162−109, 162, 227)-Net in Base 3 — Upper bound on s

There is no (53, 162, 228)-net in base 3, because

1 times m-reduction [i] would yield (53, 161, 228)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 68748 142704 361002 883627 501696 740254 202370 174953 827461 319759 239113 162775 482649 > 3¹⁶¹ [i]