MinT - Best Known (71−7, 71, s)-Nets in Base 3

Best Known (71−7, 71, s)-Nets in Base 3

(71−7, 71, 2797416)-Net over F₃ — Constructive and digital

Digital (64, 71, 2797416)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁷¹, 2797416, F₃, 9, 7) (dual of [(2797416, 9), 25176673, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(3⁷¹, 2797417, F₃, 3, 7) (dual of [(2797417, 3), 8392180, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3¹⁰, 1216, F₃, 3, 3) (dual of [(1216, 3), 3638, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(3¹⁰, 1216, F₃, 2, 3) (dual of [(1216, 2), 2422, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3¹⁰, 1216, F₃, 3) (dual of [1216, 1206, 4]-code or 1216-cap in PG(9,3)), using
        large caps in PG(u,Â 3) [i]
    2. linear OOA(3⁶¹, 2796201, F₃, 3, 7) (dual of [(2796201, 3), 8388542, 8]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(3⁶¹, large, F₃, 7) (dual of [large, large−61, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(71−7, 71, 6230222)-Net over F₃ — Digital

Digital (64, 71, 6230222)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷¹, 6230222, F₃, 7) (dual of [6230222, 6230151, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁷¹, large, F₃, 7) (dual of [large, large−71, 8]-code), using
  - 10 times code embedding in larger space [i] based on linear OA(3⁶¹, large, F₃, 7) (dual of [large, large−61, 8]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(71−7, 71, large)-Net in Base 3 — Upper bound on s

There is no (64, 71, large)-net in base 3, because

5 times m-reduction [i] would yield (64, 66, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]