Best Known (237, 237+16, s)-Nets in Base 2
(237, 237+16, 1081347)-Net over F2 — Constructive and digital
Digital (237, 253, 1081347)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (61, 69, 32772)-net over F2, using
- net defined by OOA [i] based on linear OOA(269, 32772, F2, 8, 8) (dual of [(32772, 8), 262107, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(269, 131088, F2, 8) (dual of [131088, 131019, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(269, 131090, F2, 8) (dual of [131090, 131021, 9]-code), using
- 1 times truncation [i] based on linear OA(270, 131091, F2, 9) (dual of [131091, 131021, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(269, 131072, F2, 9) (dual of [131072, 131003, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(252, 131072, F2, 7) (dual of [131072, 131020, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(218, 19, F2, 17) (dual of [19, 1, 18]-code), using
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- dual of repetition code with length 19 [i]
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- linear OA(21, 19, F2, 1) (dual of [19, 18, 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(270, 131091, F2, 9) (dual of [131091, 131021, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(269, 131090, F2, 8) (dual of [131090, 131021, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(269, 131088, F2, 8) (dual of [131088, 131019, 9]-code), using
- net defined by OOA [i] based on linear OOA(269, 32772, F2, 8, 8) (dual of [(32772, 8), 262107, 9]-NRT-code), using
- digital (168, 184, 1048575)-net over F2, using
- net defined by OOA [i] based on linear OOA(2184, 1048575, F2, 16, 16) (dual of [(1048575, 16), 16777016, 17]-NRT-code), using
- OA 8-folding and stacking [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
- OA 8-folding and stacking [i] based on linear OA(2184, 8388600, F2, 16) (dual of [8388600, 8388416, 17]-code), using
- net defined by OOA [i] based on linear OOA(2184, 1048575, F2, 16, 16) (dual of [(1048575, 16), 16777016, 17]-NRT-code), using
- digital (61, 69, 32772)-net over F2, using
(237, 237+16, 3677888)-Net over F2 — Digital
Digital (237, 253, 3677888)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2253, 3677888, F2, 2, 16) (dual of [(3677888, 2), 7355523, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2253, 4259846, F2, 2, 16) (dual of [(4259846, 2), 8519439, 17]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(269, 65545, F2, 2, 8) (dual of [(65545, 2), 131021, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(269, 131090, F2, 8) (dual of [131090, 131021, 9]-code), using
- 1 times truncation [i] based on linear OA(270, 131091, F2, 9) (dual of [131091, 131021, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(269, 131072, F2, 9) (dual of [131072, 131003, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(252, 131072, F2, 7) (dual of [131072, 131020, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(218, 19, F2, 17) (dual of [19, 1, 18]-code), using
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- dual of repetition code with length 19 [i]
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- linear OA(21, 19, F2, 1) (dual of [19, 18, 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(270, 131091, F2, 9) (dual of [131091, 131021, 10]-code), using
- OOA 2-folding [i] based on linear OA(269, 131090, F2, 8) (dual of [131090, 131021, 9]-code), using
- linear OOA(2184, 4194301, F2, 2, 16) (dual of [(4194301, 2), 8388418, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2184, 8388602, F2, 16) (dual of [8388602, 8388418, 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 2-folding [i] based on linear OA(2184, 8388602, F2, 16) (dual of [8388602, 8388418, 17]-code), using
- linear OOA(269, 65545, F2, 2, 8) (dual of [(65545, 2), 131021, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2253, 4259846, F2, 2, 16) (dual of [(4259846, 2), 8519439, 17]-NRT-code), using
(237, 237+16, large)-Net in Base 2 — Upper bound on s
There is no (237, 253, large)-net in base 2, because
- 14 times m-reduction [i] would yield (237, 239, large)-net in base 2, but