MinT - Best Known (130−83, 130, s)-Nets in Base 3

Best Known (130−83, 130, s)-Nets in Base 3

(130−83, 130, 48)-Net over F₃ — Constructive and digital

Digital (47, 130, 48)-net over F₃, using

t-expansion [i] based on digital (45, 130, 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]

(130−83, 130, 56)-Net over F₃ — Digital

Digital (47, 130, 56)-net over F₃, using

t-expansion [i] based on digital (40, 130, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(130−83, 130, 168)-Net over F₃ — Upper bound on s (digital)

There is no digital (47, 130, 169)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹³⁰, 169, F₃, 83) (dual of [169, 39, 84]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3¹²⁹, 149, S₃, 83), but
    - the linear programming bound shows that MÂ â‰¥ 84 319128 221217 735495 636798 406504 571731 768077 225953 505729 024301 422296 511468 / 1 784703 172945 > 3¹²⁹ [i]
  2. OA(3³⁹, 169, S₃, 20), but
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 4 219632 770903 631459 > 3³⁹ [i]

(130−83, 130, 217)-Net in Base 3 — Upper bound on s

There is no (47, 130, 218)-net in base 3, because

1 times m-reduction [i] would yield (47, 129, 218)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 36 155412 305309 330943 769021 491808 201666 589123 936684 825105 902245 > 3¹²⁹ [i]