MinT - Best Known (102−9, 102, s)-Nets in Base 2

Best Known (102−9, 102, s)-Nets in Base 2

(102−9, 102, 2097173)-Net over F₂ — Constructive and digital

Digital (93, 102, 2097173)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁰², 2097173, F₂, 9, 9) (dual of [(2097173, 9), 18874455, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁰², 2097173, F₂, 8, 9) (dual of [(2097173, 8), 16777282, 10]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁹, 23, F₂, 8, 4) (dual of [(23, 8), 175, 5]-NRT-code), using
      - appending 4 arbitrary columns [i] based on linear OOA(2⁹, 23, F₂, 4, 4) (dual of [(23, 4), 83, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁹, 23, F₂, 3, 4) (dual of [(23, 3), 60, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (5, 9, 23)-net over F₂, using
        linear code embedding found in a computer search [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]

(102−9, 102, 2378094)-Net over F₂ — Digital

Digital (93, 102, 2378094)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁰², 2378094, F₂, 9, 9) (dual of [(2378094, 9), 21402744, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁰², 2378094, F₂, 8, 9) (dual of [(2378094, 8), 19024650, 10]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁰², 2378094, F₂, 3, 9) (dual of [(2378094, 3), 7134180, 10]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(2¹⁰², 2796224, F₂, 3, 9) (dual of [(2796224, 3), 8388570, 10]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(2⁹, 23, F₂, 3, 4) (dual of [(23, 3), 60, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (5, 9, 23)-net over F₂, using
        linear code embedding found in a computer search [i]
        linear OOA(2⁹³, 2796201, F₂, 3, 9) (dual of [(2796201, 3), 8388510, 10]-NRT-code), using
        OOA 3-folding [i] 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]

(102−9, 102, large)-Net in Base 2 — Upper bound on s

There is no (93, 102, large)-net in base 2, because

7 times m-reduction [i] would yield (93, 95, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]