MinT - Best Known (82, 91, s)-Nets in Base 4

Best Known (82, 91, s)-Nets in Base 4

(82, 91, 2140841)-Net over F₄ — Constructive and digital

Digital (82, 91, 2140841)-net over F₄, using

4¹ times duplication [i] based on 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]

(82, 91, large)-Net over F₄ — Digital

Digital (82, 91, large)-net over F₄, using

1 times m-reduction [i] based on digital (82, 92, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁹², large, F₄, 10) (dual of [large, large−92, 11]-code), using
  - 7 times code embedding in larger space [i] based on linear OA(4⁸⁵, large, F₄, 10) (dual of [large, large−85, 11]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(82, 91, large)-Net in Base 4 — Upper bound on s

There is no (82, 91, large)-net in base 4, because

7 times m-reduction [i] would yield (82, 84, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]