MinT - Best Known (68−7, 68, s)-Nets in Base 3

Best Known (68−7, 68, s)-Nets in Base 3

(68−7, 68, 2796312)-Net over F₃ — Constructive and digital

Digital (61, 68, 2796312)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁶⁸, 2796312, F₃, 9, 7) (dual of [(2796312, 9), 25166740, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(3⁶⁸, 2796313, F₃, 3, 7) (dual of [(2796313, 3), 8388871, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3⁷, 112, F₃, 3, 3) (dual of [(112, 3), 329, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(3⁷, 112, F₃, 2, 3) (dual of [(112, 2), 217, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3⁷, 112, F₃, 3) (dual of [112, 105, 4]-code or 112-cap in PG(6,3)), using
        doubling the cap [i] based on linear OA(3⁶, 56, F₃, 3) (dual of [56, 50, 4]-code or 56-cap in PG(5,3)), using
        the Hill cap [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]

(68−7, 68, 4194413)-Net over F₃ — Digital

Digital (61, 68, 4194413)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁶⁸, 4194413, F₃, 2, 7) (dual of [(4194413, 2), 8388758, 8]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3⁷, 112, F₃, 2, 3) (dual of [(112, 2), 217, 4]-NRT-code), using
    - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3⁷, 112, F₃, 3) (dual of [112, 105, 4]-code or 112-cap in PG(6,3)), using
      - doubling the cap [i] based on linear OA(3⁶, 56, F₃, 3) (dual of [56, 50, 4]-code or 56-cap in PG(5,3)), using
        the Hill cap [i]
  2. linear OOA(3⁶¹, 4194301, F₃, 2, 7) (dual of [(4194301, 2), 8388541, 8]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3⁶¹, 8388602, F₃, 7) (dual of [8388602, 8388541, 8]-code), using
      - discarding factors / shortening the dual code 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]

(68−7, 68, large)-Net in Base 3 — Upper bound on s

There is no (61, 68, large)-net in base 3, because

5 times m-reduction [i] would yield (61, 63, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]