MinT - Best Known (66−7, 66, s)-Nets in Base 4

Best Known (66−7, 66, s)-Nets in Base 4

(66−7, 66, 2796241)-Net over F₄ — Constructive and digital

Digital (59, 66, 2796241)-net over F₄, using

net defined by OOA [i] based on linear OOA(4⁶⁶, 2796241, F₄, 9, 7) (dual of [(2796241, 9), 25166103, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(4⁶⁶, 2796242, F₄, 3, 7) (dual of [(2796242, 3), 8388660, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(4⁵, 41, F₄, 3, 3) (dual of [(41, 3), 118, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(4⁵, 41, F₄, 2, 3) (dual of [(41, 2), 77, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁵, 41, F₄, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
        a cap found by Tallinii [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]

(66−7, 66, large)-Net over F₄ — Digital

Digital (59, 66, large)-net over F₄, using

4⁵ 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]

(66−7, 66, large)-Net in Base 4 — Upper bound on s

There is no (59, 66, large)-net in base 4, because

5 times m-reduction [i] would yield (59, 61, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]