MinT - Best Known (113−9, 113, s)-Nets in Base 2

Best Known (113−9, 113, s)-Nets in Base 2

(113−9, 113, 2098175)-Net over F₂ — Constructive and digital

Digital (104, 113, 2098175)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹¹³, 2098175, F₂, 9, 9) (dual of [(2098175, 9), 18883462, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹¹³, 2098175, F₂, 8, 9) (dual of [(2098175, 8), 16785287, 10]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2²⁰, 1025, F₂, 8, 4) (dual of [(1025, 8), 8180, 5]-NRT-code), using
      - appending 4 arbitrary columns [i] based on linear OOA(2²⁰, 1025, F₂, 4, 4) (dual of [(1025, 4), 4080, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2²⁰, 1025, F₂, 3, 4) (dual of [(1025, 3), 3055, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (16, 20, 1025)-net over F₂, using
        nets constructed from net-embeddable BCH codes [i]
    2. 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]

(113−9, 113, 4194303)-Net over F₂ — Digital

Digital (104, 113, 4194303)-net over F₂, using

2¹ times duplication [i] based on digital (103, 112, 4194303)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2¹¹², 4194303, F₂, 9, 9) (dual of [(4194303, 9), 37748615, 10]-NRT-code), using
  - appending kth column [i] based on linear OOA(2¹¹², 4194303, F₂, 8, 9) (dual of [(4194303, 8), 33554312, 10]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹¹², 4194303, F₂, 3, 9) (dual of [(4194303, 3), 12582797, 10]-NRT-code), using
      - OOAs constructed from BCH codes [i]

(113−9, 113, large)-Net in Base 2 — Upper bound on s

There is no (104, 113, large)-net in base 2, because

7 times m-reduction [i] would yield (104, 106, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]