MinT - Best Known (140−22, 140, s)-Nets in Base 3

Best Known (140−22, 140, s)-Nets in Base 3

(140−22, 140, 1797)-Net over F₃ — Constructive and digital

Digital (118, 140, 1797)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
2. digital (105, 127, 1789)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹²⁷, 1789, F₃, 22, 22) (dual of [(1789, 22), 39231, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(3¹²⁷, 19679, F₃, 22) (dual of [19679, 19552, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹²⁷, 19683, F₃, 22) (dual of [19683, 19556, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(140−22, 140, 9866)-Net over F₃ — Digital

Digital (118, 140, 9866)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁴⁰, 9866, F₃, 2, 22) (dual of [(9866, 2), 19592, 23]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3¹³⁸, 9865, F₃, 2, 22) (dual of [(9865, 2), 19592, 23]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹³⁸, 19730, F₃, 22) (dual of [19730, 19592, 23]-code), using
    - constructionÂ X applied to C_e(21) âŠ‚ C_e(15) [i] based on
      1. linear OA(3¹²⁷, 19683, F₃, 22) (dual of [19683, 19556, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      2. linear OA(3⁹¹, 19683, F₃, 16) (dual of [19683, 19592, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      3. linear OA(3¹¹, 47, F₃, 5) (dual of [47, 36, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(140−22, 140, 2900246)-Net in Base 3 — Upper bound on s

There is no (118, 140, 2900247)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 265797 833209 257591 825725 855949 591721 124825 959655 858904 330976 899275 > 3¹⁴⁰ [i]