MinT - Best Known (138−87, 138, s)-Nets in Base 3

Best Known (138−87, 138, s)-Nets in Base 3

(138−87, 138, 48)-Net over F₃ — Constructive and digital

Digital (51, 138, 48)-net over F₃, using

t-expansion [i] based on digital (45, 138, 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]

(138−87, 138, 64)-Net over F₃ — Digital

Digital (51, 138, 64)-net over F₃, using

t-expansion [i] based on digital (49, 138, 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]

(138−87, 138, 233)-Net over F₃ — Upper bound on s (digital)

There is no digital (51, 138, 234)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹³⁸, 234, F₃, 87) (dual of [234, 96, 88]-code), but
- residual code [i] would yield OA(3⁵¹, 146, S₃, 29), but
  - the linear programming bound shows that MÂ â‰¥ 5126 774601 249175 366045 349858 741246 206423 340601 352320 / 2311 172455 881906 807675 878437 > 3⁵¹ [i]

(138−87, 138, 239)-Net in Base 3 — Upper bound on s

There is no (51, 138, 240)-net in base 3, because

1 times m-reduction [i] would yield (51, 137, 240)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 246471 698228 292795 404369 245973 088087 354941 259554 105285 284722 799041 > 3¹³⁷ [i]