MinT - Best Known (52, 52+105, s)-Nets in Base 3

Best Known (52, 52+105, s)-Nets in Base 3

(52, 52+105, 48)-Net over F₃ — Constructive and digital

Digital (52, 157, 48)-net over F₃, using

t-expansion [i] based on digital (45, 157, 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+105, 64)-Net over F₃ — Digital

Digital (52, 157, 64)-net over F₃, using

t-expansion [i] based on digital (49, 157, 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+105, 167)-Net over F₃ — Upper bound on s (digital)

There is no digital (52, 157, 168)-net over F₃, because

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+105, 225)-Net in Base 3 — Upper bound on s

There is no (52, 157, 226)-net in base 3, because

1 times m-reduction [i] would yield (52, 156, 226)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 313 352222 562065 758795 046923 247114 119546 422880 989980 091485 803503 557224 830729 > 3¹⁵⁶ [i]