MinT - Best Known (48, 137, s)-Nets in Base 3

Best Known (48, 137, s)-Nets in Base 3

(48, 137, 48)-Net over F₃ — Constructive and digital

Digital (48, 137, 48)-net over F₃, using

t-expansion [i] based on digital (45, 137, 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]

(48, 137, 56)-Net over F₃ — Digital

Digital (48, 137, 56)-net over F₃, using

t-expansion [i] based on digital (40, 137, 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]

(48, 137, 160)-Net over F₃ — Upper bound on s (digital)

There is no digital (48, 137, 161)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹³⁷, 161, F₃, 89) (dual of [161, 24, 90]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3¹³⁶, 149, S₃, 89), but
    - the linear programming bound shows that MÂ â‰¥ 6 385896 362423 839893 456186 755923 270794 744764 572661 578917 513381 685836 804369 / 75 686875 > 3¹³⁶ [i]
  2. OA(3²⁴, 161, S₃, 12), but
    - discarding factors would yield OA(3²⁴, 124, S₃, 12), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 293147 842993 > 3²⁴ [i]

(48, 137, 216)-Net in Base 3 — Upper bound on s

There is no (48, 137, 217)-net in base 3, because

1 times m-reduction [i] would yield (48, 136, 217)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 81116 972405 756633 000210 562654 583771 765794 884365 348159 781498 009265 > 3¹³⁶ [i]