MinT - Best Known (73−7, 73, s)-Nets in Base 3

Best Known (73−7, 73, s)-Nets in Base 3

(73−7, 73, 2802664)-Net over F₃ — Constructive and digital

Digital (66, 73, 2802664)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁷³, 2802664, F₃, 9, 7) (dual of [(2802664, 9), 25223903, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(3⁷³, 2802665, F₃, 3, 7) (dual of [(2802665, 3), 8407922, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3¹², 6464, F₃, 3, 3) (dual of [(6464, 3), 19380, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(3¹², 6464, F₃, 2, 3) (dual of [(6464, 2), 12916, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3¹², 6464, F₃, 3) (dual of [6464, 6452, 4]-code or 6464-cap in PG(11,3)), 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]

(73−7, 73, large)-Net over F₃ — Digital

Digital (66, 73, large)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷³, large, F₃, 7) (dual of [large, large−73, 8]-code), using
- 12 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]

(73−7, 73, large)-Net in Base 3 — Upper bound on s

There is no (66, 73, large)-net in base 3, because

5 times m-reduction [i] would yield (66, 68, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]