MinT - Best Known (55, 55+105, s)-Nets in Base 3

Best Known (55, 55+105, s)-Nets in Base 3

(55, 55+105, 48)-Net over F₃ — Constructive and digital

Digital (55, 160, 48)-net over F₃, using

t-expansion [i] based on digital (45, 160, 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]

(55, 55+105, 64)-Net over F₃ — Digital

Digital (55, 160, 64)-net over F₃, using

t-expansion [i] based on digital (49, 160, 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]

(55, 55+105, 192)-Net over F₃ — Upper bound on s (digital)

There is no digital (55, 160, 193)-net over F₃, because

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]

(55, 55+105, 242)-Net in Base 3 — Upper bound on s

There is no (55, 160, 243)-net in base 3, because

1 times m-reduction [i] would yield (55, 159, 243)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7442 058933 138694 782593 653075 727141 114764 013210 009142 720779 562911 523848 092537 > 3¹⁵⁹ [i]