MinT - Best Known (53, 53+90, s)-Nets in Base 3

Best Known (53, 53+90, s)-Nets in Base 3

(53, 53+90, 48)-Net over F₃ — Constructive and digital

Digital (53, 143, 48)-net over F₃, using

t-expansion [i] based on digital (45, 143, 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+90, 64)-Net over F₃ — Digital

Digital (53, 143, 64)-net over F₃, using

t-expansion [i] based on digital (49, 143, 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+90, 238)-Net over F₃ — Upper bound on s (digital)

There is no digital (53, 143, 239)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁴³, 239, F₃, 90) (dual of [239, 96, 91]-code), but
- residual code [i] would yield OA(3⁵³, 148, S₃, 30), but
  - the linear programming bound shows that MÂ â‰¥ 29 655366 158662 334400 137851 564253 047850 828558 544874 600740 / 1 414103 374055 855272 533196 393519 > 3⁵³ [i]

(53, 53+90, 247)-Net in Base 3 — Upper bound on s

There is no (53, 143, 248)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 185 169322 936687 018065 841610 082728 919920 106099 193558 173794 875162 012401 > 3¹⁴³ [i]