Best Known (243, 243+16, s)-Nets in Base 2
(243, 243+16, 1114116)-Net over F2 — Constructive and digital
Digital (243, 259, 1114116)-net over F2, using
- t-expansion [i] based on digital (242, 259, 1114116)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (66, 74, 65541)-net over F2, using
- net defined by OOA [i] based on linear OOA(274, 65541, F2, 8, 8) (dual of [(65541, 8), 524254, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(274, 262164, F2, 8) (dual of [262164, 262090, 9]-code), using
- strength reduction [i] based on linear OA(274, 262164, F2, 9) (dual of [262164, 262090, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(273, 262144, F2, 9) (dual of [262144, 262071, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(255, 262144, F2, 7) (dual of [262144, 262089, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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(274, 262164, F2, 9) (dual of [262164, 262090, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(274, 262164, F2, 8) (dual of [262164, 262090, 9]-code), using
- net defined by OOA [i] based on linear OOA(274, 65541, F2, 8, 8) (dual of [(65541, 8), 524254, 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 (66, 74, 65541)-net over F2, using
- (u, u+v)-construction [i] based on
(243, 243+16, 4325383)-Net over F2 — Digital
Digital (243, 259, 4325383)-net over F2, using
- 21 times duplication [i] based on digital (242, 258, 4325383)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 4325383, F2, 2, 16) (dual of [(4325383, 2), 8650508, 17]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(274, 131082, F2, 2, 8) (dual of [(131082, 2), 262090, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(274, 262164, F2, 8) (dual of [262164, 262090, 9]-code), using
- strength reduction [i] based on linear OA(274, 262164, F2, 9) (dual of [262164, 262090, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(273, 262144, F2, 9) (dual of [262144, 262071, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(255, 262144, F2, 7) (dual of [262144, 262089, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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(274, 262164, F2, 9) (dual of [262164, 262090, 10]-code), using
- OOA 2-folding [i] based on linear OA(274, 262164, F2, 8) (dual of [262164, 262090, 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(274, 131082, F2, 2, 8) (dual of [(131082, 2), 262090, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 4325383, F2, 2, 16) (dual of [(4325383, 2), 8650508, 17]-NRT-code), using
(243, 243+16, large)-Net in Base 2 — Upper bound on s
There is no (243, 259, large)-net in base 2, because
- 14 times m-reduction [i] would yield (243, 245, large)-net in base 2, but