MinT - Best Known (70−7, 70, s)-Nets in Base 3

Best Known (70−7, 70, s)-Nets in Base 3

(70−7, 70, 2796732)-Net over F₃ — Constructive and digital

Digital (63, 70, 2796732)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁷⁰, 2796732, F₃, 9, 7) (dual of [(2796732, 9), 25170518, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(3⁷⁰, 2796733, F₃, 3, 7) (dual of [(2796733, 3), 8390129, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3⁹, 532, F₃, 3, 3) (dual of [(532, 3), 1587, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(3⁹, 532, F₃, 2, 3) (dual of [(532, 2), 1055, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3⁹, 532, F₃, 3) (dual of [532, 523, 4]-code or 532-cap in PG(8,3)), using
        caps completed using a computer search [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]

(70−7, 70, 5001258)-Net over F₃ — Digital

Digital (63, 70, 5001258)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷⁰, 5001258, F₃, 7) (dual of [5001258, 5001188, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁷⁰, large, F₃, 7) (dual of [large, large−70, 8]-code), using
  - 9 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]

(70−7, 70, large)-Net in Base 3 — Upper bound on s

There is no (63, 70, large)-net in base 3, because

5 times m-reduction [i] would yield (63, 65, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]