MinT - Best Known (107−9, 107, s)-Nets in Base 2

Best Known (107−9, 107, s)-Nets in Base 2

(107−9, 107, 2097278)-Net over F₂ — Constructive and digital

Digital (98, 107, 2097278)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁰⁷, 2097278, F₂, 9, 9) (dual of [(2097278, 9), 18875395, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁰⁷, 2097278, F₂, 8, 9) (dual of [(2097278, 8), 16778117, 10]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2¹⁴, 128, F₂, 8, 4) (dual of [(128, 8), 1010, 5]-NRT-code), using
      - appending 4 arbitrary columns [i] based on linear OOA(2¹⁴, 128, F₂, 4, 4) (dual of [(128, 4), 498, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2¹⁴, 128, F₂, 3, 4) (dual of [(128, 3), 370, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (10, 14, 128)-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]

(107−9, 107, 2796329)-Net over F₂ — Digital

Digital (98, 107, 2796329)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁰⁷, 2796329, F₂, 9, 9) (dual of [(2796329, 9), 25166854, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁰⁷, 2796329, F₂, 8, 9) (dual of [(2796329, 8), 22370525, 10]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁰⁷, 2796329, F₂, 3, 9) (dual of [(2796329, 3), 8388880, 10]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2¹⁴, 128, F₂, 3, 4) (dual of [(128, 3), 370, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (10, 14, 128)-net over F₂, using
        linear code embedding found in a computer search [i]
      2. 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]

(107−9, 107, large)-Net in Base 2 — Upper bound on s

There is no (98, 107, large)-net in base 2, because

7 times m-reduction [i] would yield (98, 100, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]