Best Known (231, 231+16, s)-Nets in Base 2
(231, 231+16, 1056771)-Net over F2 — Constructive and digital
Digital (231, 247, 1056771)-net over F2, using
- 21 times duplication [i] based on digital (230, 246, 1056771)-net over F2, using
- t-expansion [i] based on digital (229, 246, 1056771)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (53, 61, 8196)-net over F2, using
- net defined by OOA [i] based on linear OOA(261, 8196, F2, 8, 8) (dual of [(8196, 8), 65507, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(261, 32784, F2, 8) (dual of [32784, 32723, 9]-code), using
- 1 times truncation [i] based on linear OA(262, 32785, F2, 9) (dual of [32785, 32723, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(261, 32768, F2, 9) (dual of [32768, 32707, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(246, 32768, F2, 7) (dual of [32768, 32722, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(216, 17, F2, 15) (dual of [17, 1, 16]-code), using
- strength reduction [i] based on linear OA(216, 17, F2, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,2)), using
- dual of repetition code with length 17 [i]
- strength reduction [i] based on linear OA(216, 17, F2, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,2)), using
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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(262, 32785, F2, 9) (dual of [32785, 32723, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(261, 32784, F2, 8) (dual of [32784, 32723, 9]-code), using
- net defined by OOA [i] based on linear OOA(261, 8196, F2, 8, 8) (dual of [(8196, 8), 65507, 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 (53, 61, 8196)-net over F2, using
- (u, u+v)-construction [i] based on
- t-expansion [i] based on digital (229, 246, 1056771)-net over F2, using
(231, 231+16, 2808452)-Net over F2 — Digital
Digital (231, 247, 2808452)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2247, 2808452, F2, 3, 16) (dual of [(2808452, 3), 8425109, 17]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(263, 12251, F2, 3, 8) (dual of [(12251, 3), 36690, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(263, 12251, F2, 2, 8) (dual of [(12251, 2), 24439, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(263, 16393, F2, 2, 8) (dual of [(16393, 2), 32723, 9]-NRT-code), using
- strength reduction [i] based on linear OOA(263, 16393, F2, 2, 9) (dual of [(16393, 2), 32723, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(263, 32786, F2, 9) (dual of [32786, 32723, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(262, 32785, F2, 9) (dual of [32785, 32723, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(261, 32768, F2, 9) (dual of [32768, 32707, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(246, 32768, F2, 7) (dual of [32768, 32722, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(216, 17, F2, 15) (dual of [17, 1, 16]-code), using
- strength reduction [i] based on linear OA(216, 17, F2, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,2)), using
- dual of repetition code with length 17 [i]
- strength reduction [i] based on linear OA(216, 17, F2, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,2)), using
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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 code embedding in larger space [i] based on linear OA(262, 32785, F2, 9) (dual of [32785, 32723, 10]-code), using
- OOA 2-folding [i] based on linear OA(263, 32786, F2, 9) (dual of [32786, 32723, 10]-code), using
- strength reduction [i] based on linear OOA(263, 16393, F2, 2, 9) (dual of [(16393, 2), 32723, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(263, 16393, F2, 2, 8) (dual of [(16393, 2), 32723, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(263, 12251, F2, 2, 8) (dual of [(12251, 2), 24439, 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(263, 12251, F2, 3, 8) (dual of [(12251, 3), 36690, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(231, 231+16, large)-Net in Base 2 — Upper bound on s
There is no (231, 247, large)-net in base 2, because
- 14 times m-reduction [i] would yield (231, 233, large)-net in base 2, but