MinT - Best Known (223, 223+17, s)-Nets in Base 2

Best Known (223, 223+17, s)-Nets in Base 2

(223, 223+17, 1050627)-Net over F₂ — Constructive and digital

Digital (223, 240, 1050627)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (47, 55, 2052)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2⁵⁵, 2052, F₂, 8, 8) (dual of [(2052, 8), 16361, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(2⁵⁵, 8208, F₂, 8) (dual of [8208, 8153, 9]-code), using
      - 2 times code embedding in larger space [i] based on linear OA(2⁵³, 8206, F₂, 8) (dual of [8206, 8153, 9]-code), using
        1 times truncation [i] based on linear OA(2⁵⁴, 8207, F₂, 9) (dual of [8207, 8153, 10]-code), using
        constructionÂ X4 applied to C_e(8) âŠ‚ C_e(6) [i] based on
        linear OA(2⁵³, 8192, F₂, 9) (dual of [8192, 8139, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(2⁴⁰, 8192, F₂, 7) (dual of [8192, 8152, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(2¹⁴, 15, F₂, 13) (dual of [15, 1, 14]-code), using
        strength reduction [i] based on linear OA(2¹⁴, 15, F₂, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,2)), using
        dual of repetition code with length 15 [i]
        linear OA(2¹, 15, F₂, 1) (dual of [15, 14, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹, s, F₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
2. digital (168, 185, 1048575)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹⁸⁵, 1048575, F₂, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(2¹⁸⁵, 8388601, F₂, 17) (dual of [8388601, 8388416, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁸⁵, large, F₂, 17) (dual of [large, large−185, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(223, 223+17, 2101187)-Net over F₂ — Digital

Digital (223, 240, 2101187)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁴⁰, 2101187, F₂, 4, 17) (dual of [(2101187, 4), 8404508, 18]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2⁵⁵, 4037, F₂, 4, 8) (dual of [(4037, 4), 16093, 9]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁵⁵, 4037, F₂, 2, 8) (dual of [(4037, 2), 8019, 9]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(2⁵⁵, 4104, F₂, 2, 8) (dual of [(4104, 2), 8153, 9]-NRT-code), using
        strength reduction [i] based on linear OOA(2⁵⁵, 4104, F₂, 2, 9) (dual of [(4104, 2), 8153, 10]-NRT-code), using
        OOA 2-folding [i] based on linear OA(2⁵⁵, 8208, F₂, 9) (dual of [8208, 8153, 10]-code), using
        1 times code embedding in larger space [i] based on linear OA(2⁵⁴, 8207, F₂, 9) (dual of [8207, 8153, 10]-code), using
        constructionÂ X4 applied to C_e(8) âŠ‚ C_e(6) [i] based on
        linear OA(2⁵³, 8192, F₂, 9) (dual of [8192, 8139, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(2⁴⁰, 8192, F₂, 7) (dual of [8192, 8152, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(2¹⁴, 15, F₂, 13) (dual of [15, 1, 14]-code), using
        strength reduction [i] based on linear OA(2¹⁴, 15, F₂, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,2)), using
        dual of repetition code with length 15 [i]
        linear OA(2¹, 15, F₂, 1) (dual of [15, 14, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹, s, F₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
  2. linear OOA(2¹⁸⁵, 2097150, F₂, 4, 17) (dual of [(2097150, 4), 8388415, 18]-NRT-code), using
    - OOA 4-folding [i] based on linear OA(2¹⁸⁵, 8388600, F₂, 17) (dual of [8388600, 8388415, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁸⁵, large, F₂, 17) (dual of [large, large−185, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(223, 223+17, large)-Net in Base 2 — Upper bound on s

There is no (223, 240, large)-net in base 2, because

15 times m-reduction [i] would yield (223, 225, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]

Best Known (223, 223+17, s)-Nets in Base 2

(223, 223+17, 1050627)-Net over F2 — Constructive and digital

(223, 223+17, 2101187)-Net over F2 — Digital

(223, 223+17, large)-Net in Base 2 — Upper bound on s

(223, 223+17, 1050627)-Net over F₂ — Constructive and digital

(223, 223+17, 2101187)-Net over F₂ — Digital