MinT - Best Known (140−91, 140, s)-Nets in Base 3

Best Known (140−91, 140, s)-Nets in Base 3

(140−91, 140, 48)-Net over F₃ — Constructive and digital

Digital (49, 140, 48)-net over F₃, using

t-expansion [i] based on digital (45, 140, 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]

(140−91, 140, 64)-Net over F₃ — Digital

Digital (49, 140, 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]

(140−91, 140, 181)-Net over F₃ — Upper bound on s (digital)

There is no digital (49, 140, 182)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁰, 182, F₃, 91) (dual of [182, 42, 92]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(3¹³⁹, 160, F₃, 91) (dual of [160, 21, 92]-code), but
    - constructionÂ Y1 [i] would yield
      1. OA(3¹³⁸, 150, S₃, 91), but
        the linear programming bound shows that MÂ â‰¥ 13 106423 371120 876472 986931 421169 566720 811359 868949 528224 695915 065174 686944 / 17 599301 > 3¹³⁸ [i]
      2. OA(3²¹, 160, S₃, 10), but
        discarding factors would yield OA(3²¹, 133, S₃, 10), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 10487 287539 > 3²¹ [i]
  2. OA(3⁴², 182, S₃, 22), but
    - discarding factors would yield OA(3⁴², 168, S₃, 22), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 114 464714 711551 910433 > 3⁴² [i]

(140−91, 140, 220)-Net in Base 3 — Upper bound on s

There is no (49, 140, 221)-net in base 3, because

1 times m-reduction [i] would yield (49, 139, 221)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 178049 164740 036348 013203 822153 715472 696701 392087 985473 930966 278059 > 3¹³⁹ [i]