MinT - Best Known (196−33, 196, s)-Nets in Base 3

Best Known (196−33, 196, s)-Nets in Base 3

(196−33, 196, 1480)-Net over F₃ — Constructive and digital

Digital (163, 196, 1480)-net over F₃, using

trace code for nets [i] based on digital (16, 49, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(196−33, 196, 6197)-Net over F₃ — Digital

Digital (163, 196, 6197)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹⁶, 6197, F₃, 33) (dual of [6197, 6001, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹⁶, 6629, F₃, 33) (dual of [6629, 6433, 34]-code), using
  - 3 times code embedding in larger space [i] based on linear OA(3¹⁹³, 6626, F₃, 33) (dual of [6626, 6433, 34]-code), using
    - constructionÂ X applied to C([0,16]) âŠ‚ C([0,12]) [i] based on
      1. linear OA(3¹⁷⁷, 6562, F₃, 33) (dual of [6562, 6385, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
      2. linear OA(3¹²⁹, 6562, F₃, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      3. linear OA(3¹⁶, 64, F₃, 7) (dual of [64, 48, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(196−33, 196, 2220347)-Net in Base 3 — Upper bound on s

There is no (163, 196, 2220348)-net in base 3, because

1 times m-reduction [i] would yield (163, 195, 2220348)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1093 064042 764791 914487 091325 040354 518905 781781 053414 388904 221698 163021 195970 161990 594196 305537 > 3¹⁹⁵ [i]