MinT - Best Known (181−22, 181, s)-Nets in Base 3

Best Known (181−22, 181, s)-Nets in Base 3

(181−22, 181, 48319)-Net over F₃ — Constructive and digital

Digital (159, 181, 48319)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (147, 169, 48312)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶⁹, 48312, F₃, 22, 22) (dual of [(48312, 22), 1062695, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(3¹⁶⁹, 531432, F₃, 22) (dual of [531432, 531263, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶⁹, 531441, F₃, 22) (dual of [531441, 531272, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(181−22, 181, 177167)-Net over F₃ — Digital

Digital (159, 181, 177167)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁸¹, 177167, F₃, 3, 22) (dual of [(177167, 3), 531320, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3¹⁸¹, 531501, F₃, 22) (dual of [531501, 531320, 23]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(3¹⁸⁰, 531500, F₃, 22) (dual of [531500, 531320, 23]-code), using
    - constructionÂ X applied to C_e(21) âŠ‚ C_e(15) [i] based on
      1. linear OA(3¹⁶⁹, 531441, F₃, 22) (dual of [531441, 531272, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      2. linear OA(3¹²¹, 531441, F₃, 16) (dual of [531441, 531320, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      3. linear OA(3¹¹, 59, F₃, 5) (dual of [59, 48, 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]

(181−22, 181, large)-Net in Base 3 — Upper bound on s

There is no (159, 181, large)-net in base 3, because

20 times m-reduction [i] would yield (159, 161, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]