MinT - Best Known (206−13, 206, s)-Nets in Base 2

Best Known (206−13, 206, s)-Nets in Base 2

(206−13, 206, 2796201)-Net over F₂ — Constructive and digital

Digital (193, 206, 2796201)-net over F₂, using

net defined by OOA [i] based on linear OOA(2²⁰⁶, 2796201, F₂, 18, 13) (dual of [(2796201, 18), 50331412, 14]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(2²⁰⁶, 2796202, F₂, 6, 13) (dual of [(2796202, 6), 16777006, 14]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁶⁴, 2097149, F₂, 6, 6) (dual of [(2097149, 6), 12582830, 7]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(2⁶⁴, 2097150, F₂, 3, 6) (dual of [(2097150, 3), 6291386, 7]-NRT-code), using
        1 step truncation [i] based on linear OOA(2⁶⁵, 2097151, F₂, 3, 7) (dual of [(2097151, 3), 6291388, 8]-NRT-code), using
        OOAs constructed from BCH codes [i]
    2. linear OOA(2¹⁴², 1398101, F₂, 6, 13) (dual of [(1398101, 6), 8388464, 14]-NRT-code), using
      - OOA 2-folding [i] based on linear OOA(2¹⁴², 2796202, F₂, 3, 13) (dual of [(2796202, 3), 8388464, 14]-NRT-code), using
        1 times NRT-code embedding in larger space [i] based on linear OOA(2¹³⁹, 2796201, F₂, 3, 13) (dual of [(2796201, 3), 8388464, 14]-NRT-code), using
        OOA 3-folding [i] based on linear OA(2¹³⁹, large, F₂, 13) (dual of [large, large−139, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(206−13, 206, 6265955)-Net over F₂ — Digital

Digital (193, 206, 6265955)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁰⁶, 6265955, F₂, 2, 13) (dual of [(6265955, 2), 12531704, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²⁰⁶, 8388602, F₂, 2, 13) (dual of [(8388602, 2), 16776998, 14]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁶⁷, 4194303, F₂, 2, 6) (dual of [(4194303, 2), 8388539, 7]-NRT-code), using
      - OOAs constructed from BCH codes [i]
    2. linear OOA(2¹³⁹, 4194301, F₂, 2, 13) (dual of [(4194301, 2), 8388463, 14]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(2¹³⁹, 8388602, F₂, 13) (dual of [8388602, 8388463, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹³⁹, large, F₂, 13) (dual of [large, large−139, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(206−13, 206, large)-Net in Base 2 — Upper bound on s

There is no (193, 206, large)-net in base 2, because

11 times m-reduction [i] would yield (193, 195, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]