Best Known (229, 229+19, s)-Nets in Base 2
(229, 229+19, 932199)-Net over F2 — Constructive and digital
Digital (229, 248, 932199)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (31, 40, 133)-net over F2, using
- net defined by OOA [i] based on linear OOA(240, 133, F2, 9, 9) (dual of [(133, 9), 1157, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(240, 133, F2, 8, 9) (dual of [(133, 8), 1024, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(240, 533, F2, 9) (dual of [533, 493, 10]-code), using
- adding a parity check bit [i] based on linear OA(239, 532, F2, 8) (dual of [532, 493, 9]-code), using
- construction XX applied to C1 = C([509,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([509,6]) [i] based on
- linear OA(228, 511, F2, 7) (dual of [511, 483, 8]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(227, 511, F2, 6) (dual of [511, 484, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(237, 511, F2, 9) (dual of [511, 474, 10]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(218, 511, F2, 4) (dual of [511, 493, 5]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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 (see above)
- construction XX applied to C1 = C([509,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([509,6]) [i] based on
- adding a parity check bit [i] based on linear OA(239, 532, F2, 8) (dual of [532, 493, 9]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(240, 533, F2, 9) (dual of [533, 493, 10]-code), using
- appending kth column [i] based on linear OOA(240, 133, F2, 8, 9) (dual of [(133, 8), 1024, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(240, 133, F2, 9, 9) (dual of [(133, 9), 1157, 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 (31, 40, 133)-net over F2, using
(229, 229+19, 1677954)-Net over F2 — Digital
Digital (229, 248, 1677954)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2248, 1677954, F2, 5, 19) (dual of [(1677954, 5), 8389522, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(240, 234, F2, 5, 9) (dual of [(234, 5), 1130, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(240, 234, F2, 2, 9) (dual of [(234, 2), 428, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(240, 266, F2, 2, 9) (dual of [(266, 2), 492, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(240, 532, F2, 9) (dual of [532, 492, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(240, 533, F2, 9) (dual of [533, 493, 10]-code), using
- adding a parity check bit [i] based on linear OA(239, 532, F2, 8) (dual of [532, 493, 9]-code), using
- construction XX applied to C1 = C([509,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([509,6]) [i] based on
- linear OA(228, 511, F2, 7) (dual of [511, 483, 8]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(227, 511, F2, 6) (dual of [511, 484, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(237, 511, F2, 9) (dual of [511, 474, 10]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(218, 511, F2, 4) (dual of [511, 493, 5]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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 (see above)
- construction XX applied to C1 = C([509,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([509,6]) [i] based on
- adding a parity check bit [i] based on linear OA(239, 532, F2, 8) (dual of [532, 493, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(240, 533, F2, 9) (dual of [533, 493, 10]-code), using
- OOA 2-folding [i] based on linear OA(240, 532, F2, 9) (dual of [532, 492, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(240, 266, F2, 2, 9) (dual of [(266, 2), 492, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(240, 234, F2, 2, 9) (dual of [(234, 2), 428, 10]-NRT-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(240, 234, F2, 5, 9) (dual of [(234, 5), 1130, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(229, 229+19, large)-Net in Base 2 — Upper bound on s
There is no (229, 248, large)-net in base 2, because
- 17 times m-reduction [i] would yield (229, 231, large)-net in base 2, but