MinT - Best Known (215, 231, s)-Nets in Base 2

Best Known (215, 231, s)-Nets in Base 2

(215, 231, 1049090)-Net over F₂ — Constructive and digital

Digital (215, 231, 1049090)-net over F₂, using

2¹ times duplication [i] based on digital (214, 230, 1049090)-net over F₂, using
- t-expansion [i] based on digital (213, 230, 1049090)-net over F₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (37, 45, 515)-net over F₂, using
      - net defined by OOA [i] based on linear OOA(2⁴⁵, 515, F₂, 8, 8) (dual of [(515, 8), 4075, 9]-NRT-code), using
        OA 4-folding and stacking [i] based on linear OA(2⁴⁵, 2060, F₂, 8) (dual of [2060, 2015, 9]-code), using
        1 times truncation [i] based on linear OA(2⁴⁶, 2061, F₂, 9) (dual of [2061, 2015, 10]-code), using
        constructionÂ X4 applied to C_e(8) âŠ‚ C_e(6) [i] based on
        linear OA(2⁴⁵, 2048, F₂, 9) (dual of [2048, 2003, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(2³⁴, 2048, F₂, 7) (dual of [2048, 2014, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(2¹², 13, F₂, 11) (dual of [13, 1, 12]-code), using
        strength reduction [i] based on linear OA(2¹², 13, F₂, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,2)), using
        dual of repetition code with length 13 [i]
        linear OA(2¹, 13, F₂, 1) (dual of [13, 12, 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]

(215, 231, 2772865)-Net over F₂ — Digital

Digital (215, 231, 2772865)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³¹, 2772865, F₂, 3, 16) (dual of [(2772865, 3), 8318364, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²³¹, 2796888, F₂, 3, 16) (dual of [(2796888, 3), 8390433, 17]-NRT-code), using
  - strength reduction [i] based on linear OOA(2²³¹, 2796888, F₂, 3, 17) (dual of [(2796888, 3), 8390433, 18]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2⁴⁶, 687, F₂, 3, 8) (dual of [(687, 3), 2015, 9]-NRT-code), using
        OOA 3-folding [i] based on linear OA(2⁴⁶, 2061, F₂, 8) (dual of [2061, 2015, 9]-code), using
        strength reduction [i] based on linear OA(2⁴⁶, 2061, F₂, 9) (dual of [2061, 2015, 10]-code), using
        constructionÂ X4 applied to C_e(8) âŠ‚ C_e(6) [i] based on
        linear OA(2⁴⁵, 2048, F₂, 9) (dual of [2048, 2003, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(2³⁴, 2048, F₂, 7) (dual of [2048, 2014, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(2¹², 13, F₂, 11) (dual of [13, 1, 12]-code), using
        strength reduction [i] based on linear OA(2¹², 13, F₂, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,2)), using
        dual of repetition code with length 13 [i]
        linear OA(2¹, 13, F₂, 1) (dual of [13, 12, 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, 2796201, F₂, 3, 17) (dual of [(2796201, 3), 8388418, 18]-NRT-code), using
        OOA 3-folding [i] 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]

(215, 231, large)-Net in Base 2 — Upper bound on s

There is no (215, 231, large)-net in base 2, because

14 times m-reduction [i] would yield (215, 217, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]

Best Known (215, 231, s)-Nets in Base 2

(215, 231, 1049090)-Net over F2 — Constructive and digital

(215, 231, 2772865)-Net over F2 — Digital

(215, 231, large)-Net in Base 2 — Upper bound on s

(215, 231, 1049090)-Net over F₂ — Constructive and digital

(215, 231, 2772865)-Net over F₂ — Digital