MinT - Best Known (52, 52+106, s)-Nets in Base 3

Best Known (52, 52+106, s)-Nets in Base 3

(52, 52+106, 48)-Net over F₃ — Constructive and digital

Digital (52, 158, 48)-net over F₃, using

t-expansion [i] based on digital (45, 158, 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]

(52, 52+106, 64)-Net over F₃ — Digital

Digital (52, 158, 64)-net over F₃, using

t-expansion [i] based on digital (49, 158, 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]

(52, 52+106, 167)-Net over F₃ — Upper bound on s (digital)

There is no digital (52, 158, 168)-net over F₃, because

1 times m-reduction [i] would yield digital (52, 157, 168)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁷, 168, F₃, 105) (dual of [168, 11, 106]-code), but
  - residual code [i] would yield linear OA(3⁵², 62, F₃, 35) (dual of [62, 10, 36]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(52, 52+106, 223)-Net in Base 3 — Upper bound on s

There is no (52, 158, 224)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2589 710657 504888 491428 162334 863527 334011 843149 149354 625030 315556 325000 304065 > 3¹⁵⁸ [i]