Best Known (227, 227+17, s)-Nets in Base 2
(227, 227+17, 1052675)-Net over F2 — Constructive and digital
Digital (227, 244, 1052675)-net over F2, using
- 21 times duplication [i] based on digital (226, 243, 1052675)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (50, 58, 4100)-net over F2, using
- net defined by OOA [i] based on linear OOA(258, 4100, F2, 8, 8) (dual of [(4100, 8), 32742, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(258, 16400, F2, 8) (dual of [16400, 16342, 9]-code), using
- strength reduction [i] based on linear OA(258, 16400, F2, 9) (dual of [16400, 16342, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(257, 16384, F2, 9) (dual of [16384, 16327, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(243, 16384, F2, 7) (dual of [16384, 16341, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(215, 16, F2, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,2)), using
- dual of repetition code with length 16 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- strength reduction [i] based on linear OA(258, 16400, F2, 9) (dual of [16400, 16342, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(258, 16400, F2, 8) (dual of [16400, 16342, 9]-code), using
- net defined by OOA [i] based on linear OOA(258, 4100, F2, 8, 8) (dual of [(4100, 8), 32742, 9]-NRT-code), using
- 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 8388609 | 246−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-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
- net defined by OOA [i] based on linear OOA(2185, 1048575, F2, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
- digital (50, 58, 4100)-net over F2, using
- (u, u+v)-construction [i] based on
(227, 227+17, 2157921)-Net over F2 — Digital
Digital (227, 244, 2157921)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2244, 2157921, F2, 3, 17) (dual of [(2157921, 3), 6473519, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2244, 2801668, F2, 3, 17) (dual of [(2801668, 3), 8404760, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(259, 5467, F2, 3, 8) (dual of [(5467, 3), 16342, 9]-NRT-code), using
- OOA 3-folding [i] based on linear OA(259, 16401, F2, 8) (dual of [16401, 16342, 9]-code), using
- 2 times code embedding in larger space [i] based on linear OA(257, 16399, F2, 8) (dual of [16399, 16342, 9]-code), using
- 1 times truncation [i] based on linear OA(258, 16400, F2, 9) (dual of [16400, 16342, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(257, 16384, F2, 9) (dual of [16384, 16327, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(243, 16384, F2, 7) (dual of [16384, 16341, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(215, 16, F2, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,2)), using
- dual of repetition code with length 16 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 1 times truncation [i] based on linear OA(258, 16400, F2, 9) (dual of [16400, 16342, 10]-code), using
- 2 times code embedding in larger space [i] based on linear OA(257, 16399, F2, 8) (dual of [16399, 16342, 9]-code), using
- OOA 3-folding [i] based on linear OA(259, 16401, F2, 8) (dual of [16401, 16342, 9]-code), using
- 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 8388609 | 246−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(2185, large, F2, 17) (dual of [large, large−185, 18]-code), using
- linear OOA(259, 5467, F2, 3, 8) (dual of [(5467, 3), 16342, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2244, 2801668, F2, 3, 17) (dual of [(2801668, 3), 8404760, 18]-NRT-code), using
(227, 227+17, large)-Net in Base 2 — Upper bound on s
There is no (227, 244, large)-net in base 2, because
- 15 times m-reduction [i] would yield (227, 229, large)-net in base 2, but