MinT - Best Known (85, 93, s)-Nets in Base 2

Best Known (85, 93, s)-Nets in Base 2

(85, 93, 2097150)-Net over F₂ — Constructive and digital

Digital (85, 93, 2097150)-net over F₂, using

t-expansion [i] based on digital (84, 93, 2097150)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2⁹³, 2097150, F₂, 9, 9) (dual of [(2097150, 9), 18874257, 10]-NRT-code), using
  - appending kth column [i] based on linear OOA(2⁹³, 2097150, F₂, 8, 9) (dual of [(2097150, 8), 16777107, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(2⁹³, 8388601, F₂, 9) (dual of [8388601, 8388508, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(2⁹³, large, F₂, 9) (dual of [large, large−93, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(85, 93, 2796201)-Net over F₂ — Digital

Digital (85, 93, 2796201)-net over F₂, using

2¹ times duplication [i] based on digital (84, 92, 2796201)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2⁹², 2796201, F₂, 8, 8) (dual of [(2796201, 8), 22369516, 9]-NRT-code), using
  - appending kth column [i] based on linear OOA(2⁹², 2796201, F₂, 7, 8) (dual of [(2796201, 7), 19573315, 9]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁹², 2796201, F₂, 3, 8) (dual of [(2796201, 3), 8388511, 9]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(2⁹², large, F₂, 8) (dual of [large, large−92, 9]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(85, 93, large)-Net in Base 2 — Upper bound on s

There is no (85, 93, large)-net in base 2, because

6 times m-reduction [i] would yield (85, 87, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]