Best Known (218−16, 218, s)-Nets in Base 2
(218−16, 218, 1048653)-Net over F2 — Constructive and digital
Digital (202, 218, 1048653)-net over F2, using
- 21 times duplication [i] based on digital (201, 217, 1048653)-net over F2, using
- t-expansion [i] based on digital (200, 217, 1048653)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (24, 32, 78)-net over F2, using
- net defined by OOA [i] based on linear OOA(232, 78, F2, 8, 8) (dual of [(78, 8), 592, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(232, 78, F2, 7, 8) (dual of [(78, 7), 514, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(211, 47, F2, 7, 4) (dual of [(47, 7), 318, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(211, 47, F2, 4, 4) (dual of [(47, 4), 177, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(211, 47, F2, 3, 4) (dual of [(47, 3), 130, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (7, 11, 47)-net over F2, using
- appending kth column [i] based on linear OOA(211, 47, F2, 3, 4) (dual of [(47, 3), 130, 5]-NRT-code), using
- appending 3 arbitrary columns [i] based on linear OOA(211, 47, F2, 4, 4) (dual of [(47, 4), 177, 5]-NRT-code), using
- linear OOA(221, 39, F2, 7, 8) (dual of [(39, 7), 252, 9]-NRT-code), using
- extracting embedded OOA [i] based on digital (13, 21, 39)-net over F2, using
- linear OOA(211, 47, F2, 7, 4) (dual of [(47, 7), 318, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(232, 78, F2, 7, 8) (dual of [(78, 7), 514, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(232, 78, F2, 8, 8) (dual of [(78, 8), 592, 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 (24, 32, 78)-net over F2, using
- (u, u+v)-construction [i] based on
- t-expansion [i] based on digital (200, 217, 1048653)-net over F2, using
(218−16, 218, 2097286)-Net over F2 — Digital
Digital (202, 218, 2097286)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2218, 2097286, F2, 4, 16) (dual of [(2097286, 4), 8388926, 17]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(234, 136, F2, 4, 8) (dual of [(136, 4), 510, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(234, 136, F2, 2, 8) (dual of [(136, 2), 238, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(234, 272, F2, 8) (dual of [272, 238, 9]-code), using
- 1 times truncation [i] based on linear OA(235, 273, F2, 9) (dual of [273, 238, 10]-code), using
- construction XX applied to C1 = C([253,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- linear OA(225, 255, F2, 7) (dual of [255, 230, 8]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(225, 255, F2, 7) (dual of [255, 230, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(233, 255, F2, 9) (dual of [255, 222, 10]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(217, 255, F2, 5) (dual of [255, 238, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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
- linear OA(21, 9, F2, 1) (dual of [9, 8, 2]-code) (see above)
- construction XX applied to C1 = C([253,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- 1 times truncation [i] based on linear OA(235, 273, F2, 9) (dual of [273, 238, 10]-code), using
- OOA 2-folding [i] based on linear OA(234, 272, F2, 8) (dual of [272, 238, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(234, 136, F2, 2, 8) (dual of [(136, 2), 238, 9]-NRT-code), using
- 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 8388607 = 223−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(2184, large, F2, 16) (dual of [large, large−184, 17]-code), using
- OOA 4-folding [i] based on linear OA(2184, 8388600, F2, 16) (dual of [8388600, 8388416, 17]-code), using
- linear OOA(234, 136, F2, 4, 8) (dual of [(136, 4), 510, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(218−16, 218, large)-Net in Base 2 — Upper bound on s
There is no (202, 218, large)-net in base 2, because
- 14 times m-reduction [i] would yield (202, 204, large)-net in base 2, but