MinT - Best Known (53, 145, s)-Nets in Base 3

Best Known (53, 145, s)-Nets in Base 3

(53, 145, 48)-Net over F₃ — Constructive and digital

Digital (53, 145, 48)-net over F₃, using

t-expansion [i] based on digital (45, 145, 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, 145, 64)-Net over F₃ — Digital

Digital (53, 145, 64)-net over F₃, using

t-expansion [i] based on digital (49, 145, 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, 145, 238)-Net over F₃ — Upper bound on s (digital)

There is no digital (53, 145, 239)-net over F₃, because

2 times m-reduction [i] would yield digital (53, 143, 239)-net over F₃, but
- 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, 145, 244)-Net in Base 3 — Upper bound on s

There is no (53, 145, 245)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1661 304172 123144 797428 162585 224341 518515 741735 814134 431558 735871 095873 > 3¹⁴⁵ [i]