MinT - Best Known (81, 81+9, s)-Nets in Base 4

Best Known (81, 81+9, s)-Nets in Base 4

(81, 81+9, 2140841)-Net over F₄ — Constructive and digital

Digital (81, 90, 2140841)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 17, 43691)-net over F₄, using
  - nets constructed from net-embeddable BCH codes [i]
2. digital (64, 73, 2097150)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁷³, 2097150, F₄, 9, 9) (dual of [(2097150, 9), 18874277, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(4⁷³, 8388601, F₄, 9) (dual of [8388601, 8388528, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁷³, large, F₄, 9) (dual of [large, large−73, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(81, 81+9, large)-Net over F₄ — Digital

Digital (81, 90, large)-net over F₄, using

4⁹ times duplication [i] based on digital (72, 81, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁸¹, large, F₄, 9) (dual of [large, large−81, 10]-code), using
  - 8 times code embedding in larger space [i] based on linear OA(4⁷³, large, F₄, 9) (dual of [large, large−73, 10]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(81, 81+9, large)-Net in Base 4 — Upper bound on s

There is no (81, 90, large)-net in base 4, because

7 times m-reduction [i] would yield (81, 83, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]