MinT - Best Known (162−107, 162, s)-Nets in Base 3

Best Known (162−107, 162, s)-Nets in Base 3

(162−107, 162, 48)-Net over F₃ — Constructive and digital

Digital (55, 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]

(162−107, 162, 64)-Net over F₃ — Digital

Digital (55, 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]

(162−107, 162, 192)-Net over F₃ — Upper bound on s (digital)

There is no digital (55, 162, 193)-net over F₃, because

2 times m-reduction [i] would yield digital (55, 160, 193)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁶⁰, 193, F₃, 105) (dual of [193, 33, 106]-code), but
  - residual code [i] would yield OA(3⁵⁵, 87, S₃, 35), but
    - the linear programming bound shows that MÂ â‰¥ 4 080243 847169 139626 292321 742040 788787 487274 687791 / 20316 463038 499354 974208 > 3⁵⁵ [i]

(162−107, 162, 240)-Net in Base 3 — Upper bound on s

There is no (55, 162, 241)-net in base 3, because

1 times m-reduction [i] would yield (55, 161, 241)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 66345 200044 865108 325467 080974 508948 971379 394149 357101 549440 490155 936531 936787 > 3¹⁶¹ [i]