Best Known (242−18, 242, s)-Nets in Base 2
(242−18, 242, 932135)-Net over F2 — Constructive and digital
Digital (224, 242, 932135)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (26, 35, 68)-net over F2, using
- net defined by OOA [i] based on linear OOA(235, 68, F2, 9, 9) (dual of [(68, 9), 577, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(235, 68, F2, 8, 9) (dual of [(68, 8), 509, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(235, 273, F2, 9) (dual of [273, 238, 10]-code), using
- construction XX applied to C1 = C([253,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- linear OA(225, 255, F2, 7) (dual of [255, 230, 8]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(225, 255, F2, 7) (dual of [255, 230, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(233, 255, F2, 9) (dual of [255, 222, 10]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(217, 255, F2, 5) (dual of [255, 238, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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, 9, F2, 1) (dual of [9, 8, 2]-code) (see above)
- construction XX applied to C1 = C([253,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(235, 273, F2, 9) (dual of [273, 238, 10]-code), using
- appending kth column [i] based on linear OOA(235, 68, F2, 8, 9) (dual of [(68, 8), 509, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(235, 68, F2, 9, 9) (dual of [(68, 9), 577, 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 (26, 35, 68)-net over F2, using
(242−18, 242, 1838935)-Net over F2 — Digital
Digital (224, 242, 1838935)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2242, 1838935, F2, 4, 18) (dual of [(1838935, 4), 7355498, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2242, 2097218, F2, 4, 18) (dual of [(2097218, 4), 8388630, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OOA(2242, 4194436, F2, 2, 18) (dual of [(4194436, 2), 8388630, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2242, 4194437, F2, 2, 18) (dual of [(4194437, 2), 8388632, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(235, 136, F2, 2, 9) (dual of [(136, 2), 237, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(235, 272, F2, 9) (dual of [272, 237, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(235, 273, F2, 9) (dual of [273, 238, 10]-code), using
- construction XX applied to C1 = C([253,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- linear OA(225, 255, F2, 7) (dual of [255, 230, 8]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(225, 255, F2, 7) (dual of [255, 230, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(233, 255, F2, 9) (dual of [255, 222, 10]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(217, 255, F2, 5) (dual of [255, 238, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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, 9, F2, 1) (dual of [9, 8, 2]-code) (see above)
- construction XX applied to C1 = C([253,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(235, 273, F2, 9) (dual of [273, 238, 10]-code), using
- OOA 2-folding [i] based on linear OA(235, 272, F2, 9) (dual of [272, 237, 10]-code), using
- linear OOA(2207, 4194301, F2, 2, 18) (dual of [(4194301, 2), 8388395, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2207, 8388602, F2, 18) (dual of [8388602, 8388395, 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 2-folding [i] based on linear OA(2207, 8388602, F2, 18) (dual of [8388602, 8388395, 19]-code), using
- linear OOA(235, 136, F2, 2, 9) (dual of [(136, 2), 237, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2242, 4194437, F2, 2, 18) (dual of [(4194437, 2), 8388632, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OOA(2242, 4194436, F2, 2, 18) (dual of [(4194436, 2), 8388630, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2242, 2097218, F2, 4, 18) (dual of [(2097218, 4), 8388630, 19]-NRT-code), using
(242−18, 242, large)-Net in Base 2 — Upper bound on s
There is no (224, 242, large)-net in base 2, because
- 16 times m-reduction [i] would yield (224, 226, large)-net in base 2, but