MinT - Best Known (212−16, 212, s)-Nets in Base 2

Best Known (212−16, 212, s)-Nets in Base 2

(212−16, 212, 1048628)-Net over F₂ — Constructive and digital

Digital (196, 212, 1048628)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (20, 28, 53)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2²⁸, 53, F₂, 8, 8) (dual of [(53, 8), 396, 9]-NRT-code), using
    - appending kth column [i] based on linear OOA(2²⁸, 53, F₂, 7, 8) (dual of [(53, 7), 343, 9]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(2⁹, 23, F₂, 7, 4) (dual of [(23, 7), 152, 5]-NRT-code), using
        appending 3 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]
        linear OOA(2¹⁹, 30, F₂, 7, 8) (dual of [(30, 7), 191, 9]-NRT-code), using
        extracting embedded OOA [i] based on digital (11, 19, 30)-net over F₂, using
        linear code embedding found in a computer search [i]
2. digital (168, 184, 1048575)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹⁸⁴, 1048575, F₂, 16, 16) (dual of [(1048575, 16), 16777016, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(2¹⁸⁴, 8388600, F₂, 16) (dual of [8388600, 8388416, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁸⁴, large, F₂, 16) (dual of [large, large−184, 17]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(212−16, 212, 2097214)-Net over F₂ — Digital

Digital (196, 212, 2097214)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²¹², 2097214, F₂, 4, 16) (dual of [(2097214, 4), 8388644, 17]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2²⁸, 64, F₂, 4, 8) (dual of [(64, 4), 228, 9]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁸, 64, F₂, 2, 8) (dual of [(64, 2), 100, 9]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(2²⁸, 128, F₂, 8) (dual of [128, 100, 9]-code), using
        1 times truncation [i] based on linear OA(2²⁹, 129, F₂, 9) (dual of [129, 100, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 129Â | 2¹⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
  2. linear OOA(2¹⁸⁴, 2097150, F₂, 4, 16) (dual of [(2097150, 4), 8388416, 17]-NRT-code), using
    - OOA 4-folding [i] based on linear OA(2¹⁸⁴, 8388600, F₂, 16) (dual of [8388600, 8388416, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁸⁴, large, F₂, 16) (dual of [large, large−184, 17]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(212−16, 212, large)-Net in Base 2 — Upper bound on s

There is no (196, 212, large)-net in base 2, because

14 times m-reduction [i] would yield (196, 198, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]

Best Known (212−16, 212, s)-Nets in Base 2

(212−16, 212, 1048628)-Net over F2 — Constructive and digital

(212−16, 212, 2097214)-Net over F2 — Digital

(212−16, 212, large)-Net in Base 2 — Upper bound on s

(212−16, 212, 1048628)-Net over F₂ — Constructive and digital

(212−16, 212, 2097214)-Net over F₂ — Digital