Best Known (226, 243, s)-Nets in Base 2
(226, 243, 1052675)-Net over F2 — Constructive and digital
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
(226, 243, 2103272)-Net over F2 — Digital
Digital (226, 243, 2103272)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2243, 2103272, F2, 4, 17) (dual of [(2103272, 4), 8412845, 18]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(258, 6122, F2, 4, 8) (dual of [(6122, 4), 24430, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(258, 6122, F2, 2, 8) (dual of [(6122, 2), 12186, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(258, 8200, F2, 2, 8) (dual of [(8200, 2), 16342, 9]-NRT-code), using
- OOA 2-folding [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
- OOA 2-folding [i] based on linear OA(258, 16400, F2, 8) (dual of [16400, 16342, 9]-code), using
- discarding factors / shortening the dual code based on linear OOA(258, 8200, F2, 2, 8) (dual of [(8200, 2), 16342, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(258, 6122, F2, 2, 8) (dual of [(6122, 2), 12186, 9]-NRT-code), using
- 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 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 4-folding [i] based on linear OA(2185, 8388600, F2, 17) (dual of [8388600, 8388415, 18]-code), using
- linear OOA(258, 6122, F2, 4, 8) (dual of [(6122, 4), 24430, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(226, 243, large)-Net in Base 2 — Upper bound on s
There is no (226, 243, large)-net in base 2, because
- 15 times m-reduction [i] would yield (226, 228, large)-net in base 2, but