MinT - Best Known (70−7, 70, s)-Nets in Base 4

Best Known (70−7, 70, s)-Nets in Base 4

(70−7, 70, 2798310)-Net over F₄ — Constructive and digital

Digital (63, 70, 2798310)-net over F₄, using

net defined by OOA [i] based on linear OOA(4⁷⁰, 2798310, F₄, 9, 7) (dual of [(2798310, 9), 25184720, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(4⁷⁰, 2798311, F₄, 3, 7) (dual of [(2798311, 3), 8394863, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(4⁹, 2110, F₄, 3, 3) (dual of [(2110, 3), 6321, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(4⁹, 2110, F₄, 2, 3) (dual of [(2110, 2), 4211, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁹, 2110, F₄, 3) (dual of [2110, 2101, 4]-code or 2110-cap in PG(8,4)), using
        caps completed using a computer search [i]
    2. linear OOA(4⁶¹, 2796201, F₄, 3, 7) (dual of [(2796201, 3), 8388542, 8]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(4⁶¹, large, F₄, 7) (dual of [large, large−61, 8]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(70−7, 70, large)-Net over F₄ — Digital

Digital (63, 70, large)-net over F₄, using

4⁹ times duplication [i] based on digital (54, 61, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁶¹, large, F₄, 7) (dual of [large, large−61, 8]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(70−7, 70, large)-Net in Base 4 — Upper bound on s

There is no (63, 70, large)-net in base 4, because

5 times m-reduction [i] would yield (63, 65, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]