Best Known (219, 219+16, s)-Nets in Base 2
(219, 219+16, 1049602)-Net over F2 — Constructive and digital
Digital (219, 235, 1049602)-net over F2, using
- 21 times duplication [i] based on digital (218, 234, 1049602)-net over F2, using
- t-expansion [i] based on digital (217, 234, 1049602)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (41, 49, 1027)-net over F2, using
- net defined by OOA [i] based on linear OOA(249, 1027, F2, 8, 8) (dual of [(1027, 8), 8167, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(249, 4108, F2, 8) (dual of [4108, 4059, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(249, 4109, F2, 8) (dual of [4109, 4060, 9]-code), using
- 1 times truncation [i] based on linear OA(250, 4110, F2, 9) (dual of [4110, 4060, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(249, 4096, F2, 9) (dual of [4096, 4047, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(237, 4096, F2, 7) (dual of [4096, 4059, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(213, 14, F2, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,2)), using
- dual of repetition code with length 14 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(250, 4110, F2, 9) (dual of [4110, 4060, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(249, 4109, F2, 8) (dual of [4109, 4060, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(249, 4108, F2, 8) (dual of [4108, 4059, 9]-code), using
- net defined by OOA [i] based on linear OOA(249, 1027, F2, 8, 8) (dual of [(1027, 8), 8167, 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 (41, 49, 1027)-net over F2, using
- (u, u+v)-construction [i] based on
- t-expansion [i] based on digital (217, 234, 1049602)-net over F2, using
(219, 219+16, 2798256)-Net over F2 — Digital
Digital (219, 235, 2798256)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2235, 2798256, F2, 3, 16) (dual of [(2798256, 3), 8394533, 17]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(251, 2055, F2, 3, 8) (dual of [(2055, 3), 6114, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(251, 2055, F2, 2, 8) (dual of [(2055, 2), 4059, 9]-NRT-code), using
- 21 times duplication [i] based on linear OOA(250, 2055, F2, 2, 8) (dual of [(2055, 2), 4060, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(250, 4110, F2, 8) (dual of [4110, 4060, 9]-code), using
- strength reduction [i] based on linear OA(250, 4110, F2, 9) (dual of [4110, 4060, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(249, 4096, F2, 9) (dual of [4096, 4047, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(237, 4096, F2, 7) (dual of [4096, 4059, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(213, 14, F2, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,2)), using
- dual of repetition code with length 14 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(250, 4110, F2, 9) (dual of [4110, 4060, 10]-code), using
- OOA 2-folding [i] based on linear OA(250, 4110, F2, 8) (dual of [4110, 4060, 9]-code), using
- 21 times duplication [i] based on linear OOA(250, 2055, F2, 2, 8) (dual of [(2055, 2), 4060, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(251, 2055, F2, 2, 8) (dual of [(2055, 2), 4059, 9]-NRT-code), using
- linear OOA(2184, 2796201, F2, 3, 16) (dual of [(2796201, 3), 8388419, 17]-NRT-code), using
- OOA 3-folding [i] 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]
- OOA 3-folding [i] based on linear OA(2184, large, F2, 16) (dual of [large, large−184, 17]-code), using
- linear OOA(251, 2055, F2, 3, 8) (dual of [(2055, 3), 6114, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(219, 219+16, large)-Net in Base 2 — Upper bound on s
There is no (219, 235, large)-net in base 2, because
- 14 times m-reduction [i] would yield (219, 221, large)-net in base 2, but