Best Known (229, 229+18, s)-Nets in Base 2
(229, 229+18, 932200)-Net over F2 — Constructive and digital
Digital (229, 247, 932200)-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, 207, 932067)-net over F2, using
- net defined by OOA [i] based on linear OOA(2207, 932067, F2, 18, 18) (dual of [(932067, 18), 16776999, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- net defined by OOA [i] based on linear OOA(2207, 932067, F2, 18, 18) (dual of [(932067, 18), 16776999, 19]-NRT-code), using
- digital (31, 40, 133)-net over F2, using
(229, 229+18, 2097384)-Net over F2 — Digital
Digital (229, 247, 2097384)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2247, 2097384, F2, 4, 18) (dual of [(2097384, 4), 8389289, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(240, 234, F2, 4, 9) (dual of [(234, 4), 896, 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(2207, 2097150, F2, 4, 18) (dual of [(2097150, 4), 8388393, 19]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- OOA 4-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-code), using
- linear OOA(240, 234, F2, 4, 9) (dual of [(234, 4), 896, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(229, 229+18, large)-Net in Base 2 — Upper bound on s
There is no (229, 247, large)-net in base 2, because
- 16 times m-reduction [i] would yield (229, 231, large)-net in base 2, but