Best Known (238−18, 238, s)-Nets in Base 2
(238−18, 238, 932125)-Net over F2 — Constructive and digital
Digital (220, 238, 932125)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (22, 31, 58)-net over F2, using
- net defined by OOA [i] based on linear OOA(231, 58, F2, 9, 9) (dual of [(58, 9), 491, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(231, 58, F2, 8, 9) (dual of [(58, 8), 433, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(210, 32, F2, 8, 4) (dual of [(32, 8), 246, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 10, 32)-net over F2, using
- appending kth column [i] based on linear OOA(210, 32, F2, 3, 4) (dual of [(32, 3), 86, 5]-NRT-code), using
- appending 4 arbitrary columns [i] based on linear OOA(210, 32, F2, 4, 4) (dual of [(32, 4), 118, 5]-NRT-code), using
- linear OOA(221, 29, F2, 8, 9) (dual of [(29, 8), 211, 10]-NRT-code), using
- extracting embedded OOA [i] based on digital (12, 21, 29)-net over F2, using
- linear OOA(210, 32, F2, 8, 4) (dual of [(32, 8), 246, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(231, 58, F2, 8, 9) (dual of [(58, 8), 433, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(231, 58, F2, 9, 9) (dual of [(58, 9), 491, 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 (22, 31, 58)-net over F2, using
(238−18, 238, 1677791)-Net over F2 — Digital
Digital (220, 238, 1677791)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2238, 1677791, F2, 5, 18) (dual of [(1677791, 5), 8388717, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(231, 71, F2, 5, 9) (dual of [(71, 5), 324, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(231, 71, F2, 2, 9) (dual of [(71, 2), 111, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(231, 142, F2, 9) (dual of [142, 111, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(231, 143, F2, 9) (dual of [143, 112, 10]-code), using
- construction XX applied to C1 = C({0,1,3,63}), C2 = C([0,5]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C({0,1,3,5,63}) [i] based on
- linear OA(222, 127, F2, 7) (dual of [127, 105, 8]-code), using the primitive cyclic code C(A) with length 127 = 27−1, defining set A = {0,1,3,63}, and minimum distance d ≥ |{−2,−1,…,4}|+1 = 8 (BCH-bound) [i]
- linear OA(222, 127, F2, 7) (dual of [127, 105, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 127 = 27−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(229, 127, F2, 9) (dual of [127, 98, 10]-code), using the primitive cyclic code C(A) with length 127 = 27−1, defining set A = {0,1,3,5,63}, and minimum distance d ≥ |{−2,−1,…,6}|+1 = 10 (BCH-bound) [i]
- linear OA(215, 127, F2, 5) (dual of [127, 112, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 127 = 27−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(21, 8, F2, 1) (dual of [8, 7, 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, 8, F2, 1) (dual of [8, 7, 2]-code) (see above)
- construction XX applied to C1 = C({0,1,3,63}), C2 = C([0,5]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C({0,1,3,5,63}) [i] based on
- discarding factors / shortening the dual code based on linear OA(231, 143, F2, 9) (dual of [143, 112, 10]-code), using
- OOA 2-folding [i] based on linear OA(231, 142, F2, 9) (dual of [142, 111, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(231, 71, F2, 2, 9) (dual of [(71, 2), 111, 10]-NRT-code), using
- linear OOA(2207, 1677720, F2, 5, 18) (dual of [(1677720, 5), 8388393, 19]-NRT-code), using
- OOA 5-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 5-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-code), using
- linear OOA(231, 71, F2, 5, 9) (dual of [(71, 5), 324, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(238−18, 238, large)-Net in Base 2 — Upper bound on s
There is no (220, 238, large)-net in base 2, because
- 16 times m-reduction [i] would yield (220, 222, large)-net in base 2, but