MinT - Best Known (191−19, 191, s)-Nets in Base 3

Best Known (191−19, 191, s)-Nets in Base 3

(191−19, 191, 932073)-Net over F₃ — Constructive and digital

Digital (172, 191, 932073)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 10, 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 (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]

(191−19, 191, 2796208)-Net over F₃ — Digital

Digital (172, 191, 2796208)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁹¹, 2796208, F₃, 3, 19) (dual of [(2796208, 3), 8388433, 20]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁰, 7, F₃, 3, 9) (dual of [(7, 3), 11, 10]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,11P) [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [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]

(191−19, 191, large)-Net in Base 3 — Upper bound on s

There is no (172, 191, large)-net in base 3, because

17 times m-reduction [i] would yield (172, 174, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]