MinT - Best Known (245−17, 245, s)-Nets in Base 2

Best Known (245−17, 245, s)-Nets in Base 2

(245−17, 245, 1056767)-Net over F₂ — Constructive and digital

Digital (228, 245, 1056767)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (52, 60, 8192)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2⁶⁰, 8192, F₂, 8, 8) (dual of [(8192, 8), 65476, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(2⁶⁰, 32768, F₂, 8) (dual of [32768, 32708, 9]-code), using
      - 1 times truncation [i] based on linear OA(2⁶¹, 32769, F₂, 9) (dual of [32769, 32708, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 32769Â | 2³⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
2. digital (168, 185, 1048575)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹⁸⁵, 1048575, F₂, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(2¹⁸⁵, 8388601, F₂, 17) (dual of [8388601, 8388416, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁸⁵, large, F₂, 17) (dual of [large, large−185, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(245−17, 245, 2276102)-Net over F₂ — Digital

Digital (228, 245, 2276102)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁴⁵, 2276102, F₂, 3, 17) (dual of [(2276102, 3), 6828061, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²⁴⁵, 2807123, F₂, 3, 17) (dual of [(2807123, 3), 8421124, 18]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁶⁰, 10922, F₂, 3, 8) (dual of [(10922, 3), 32706, 9]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(2⁶⁰, 32766, F₂, 8) (dual of [32766, 32706, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(2⁶⁰, 32768, F₂, 8) (dual of [32768, 32708, 9]-code), using
        1 times truncation [i] based on linear OA(2⁶¹, 32769, F₂, 9) (dual of [32769, 32708, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 32769Â | 2³⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
    2. linear OOA(2¹⁸⁵, 2796201, F₂, 3, 17) (dual of [(2796201, 3), 8388418, 18]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(2¹⁸⁵, large, F₂, 17) (dual of [large, large−185, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(245−17, 245, large)-Net in Base 2 — Upper bound on s

There is no (228, 245, large)-net in base 2, because

15 times m-reduction [i] would yield (228, 230, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]