Best Known (222, 241, s)-Nets in Base 2
(222, 241, 932130)-Net over F2 — Constructive and digital
Digital (222, 241, 932130)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (24, 33, 64)-net over F2, using
- net defined by OOA [i] based on linear OOA(233, 64, F2, 9, 9) (dual of [(64, 9), 543, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(233, 64, F2, 8, 9) (dual of [(64, 8), 479, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(233, 257, F2, 9) (dual of [257, 224, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 257 | 216−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(233, 257, F2, 9) (dual of [257, 224, 10]-code), using
- appending kth column [i] based on linear OOA(233, 64, F2, 8, 9) (dual of [(64, 8), 479, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(233, 64, F2, 9, 9) (dual of [(64, 9), 543, 10]-NRT-code), using
- digital (189, 208, 932066)-net over F2, using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
- digital (24, 33, 64)-net over F2, using
(222, 241, 1652927)-Net over F2 — Digital
Digital (222, 241, 1652927)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2241, 1652927, F2, 5, 19) (dual of [(1652927, 5), 8264394, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2241, 1677784, F2, 5, 19) (dual of [(1677784, 5), 8388679, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(233, 64, F2, 5, 9) (dual of [(64, 5), 287, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(233, 257, F2, 9) (dual of [257, 224, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 257 | 216−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(233, 257, F2, 9) (dual of [257, 224, 10]-code), using
- linear OOA(2208, 1677720, F2, 5, 19) (dual of [(1677720, 5), 8388392, 20]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2208, 8388600, F2, 19) (dual of [8388600, 8388392, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 5-folding [i] based on linear OA(2208, 8388600, F2, 19) (dual of [8388600, 8388392, 20]-code), using
- linear OOA(233, 64, F2, 5, 9) (dual of [(64, 5), 287, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2241, 1677784, F2, 5, 19) (dual of [(1677784, 5), 8388679, 20]-NRT-code), using
(222, 241, large)-Net in Base 2 — Upper bound on s
There is no (222, 241, large)-net in base 2, because
- 17 times m-reduction [i] would yield (222, 224, large)-net in base 2, but