MinT - Best Known (195−19, 195, s)-Nets in Base 3

Best Known (195−19, 195, s)-Nets in Base 3

(195−19, 195, 932081)-Net over F₃ — Constructive and digital

Digital (176, 195, 932081)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (5, 14, 15)-net over F₃, using
  - linear code embedding found in a computer search [i]
2. digital (162, 181, 932066)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁸¹, 932066, F₃, 19, 19) (dual of [(932066, 19), 17709073, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(3¹⁸¹, 8388595, F₃, 19) (dual of [8388595, 8388414, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁸¹, large, F₃, 19) (dual of [large, large−181, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(195−19, 195, 2796216)-Net over F₃ — Digital

Digital (176, 195, 2796216)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁹⁵, 2796216, F₃, 3, 19) (dual of [(2796216, 3), 8388453, 20]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁴, 15, F₃, 3, 9) (dual of [(15, 3), 31, 10]-NRT-code), using
    - extracting embedded OOA [i] based on digital (5, 14, 15)-net over F₃, using
      - linear code embedding found in a computer search [i]
  2. linear OOA(3¹⁸¹, 2796201, F₃, 3, 19) (dual of [(2796201, 3), 8388422, 20]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(3¹⁸¹, large, F₃, 19) (dual of [large, large−181, 20]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(195−19, 195, large)-Net in Base 3 — Upper bound on s

There is no (176, 195, large)-net in base 3, because

17 times m-reduction [i] would yield (176, 178, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]