MinT - Best Known (51, 51+105, s)-Nets in Base 3

Best Known (51, 51+105, s)-Nets in Base 3

(51, 51+105, 48)-Net over F₃ — Constructive and digital

Digital (51, 156, 48)-net over F₃, using

t-expansion [i] based on digital (45, 156, 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]

(51, 51+105, 64)-Net over F₃ — Digital

Digital (51, 156, 64)-net over F₃, using

t-expansion [i] based on digital (49, 156, 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]

(51, 51+105, 163)-Net over F₃ — Upper bound on s (digital)

There is no digital (51, 156, 164)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁶, 164, F₃, 105) (dual of [164, 8, 106]-code), but
- residual code [i] would yield OA(3⁵¹, 58, S₃, 35), but
  - the linear programming bound shows that MÂ â‰¥ 2151 540269 112482 208544 436253 / 946 > 3⁵¹ [i]

(51, 51+105, 219)-Net in Base 3 — Upper bound on s

There is no (51, 156, 220)-net in base 3, because

1 times m-reduction [i] would yield (51, 155, 220)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 97 566970 545879 617895 288724 978711 603928 315342 372511 872711 863861 734743 578497 > 3¹⁵⁵ [i]