Best Known (252−19, 252, s)-Nets in Base 2
(252−19, 252, 932327)-Net over F2 — Constructive and digital
Digital (233, 252, 932327)-net over F2, using
- 21 times duplication [i] based on digital (232, 251, 932327)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (34, 43, 261)-net over F2, using
- net defined by OOA [i] based on linear OOA(243, 261, F2, 9, 9) (dual of [(261, 9), 2306, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(243, 261, F2, 8, 9) (dual of [(261, 8), 2045, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(243, 1045, F2, 9) (dual of [1045, 1002, 10]-code), using
- construction XX applied to C1 = C([1021,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- linear OA(231, 1023, F2, 7) (dual of [1023, 992, 8]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(231, 1023, F2, 7) (dual of [1023, 992, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(221, 1023, F2, 5) (dual of [1023, 1002, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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, 11, F2, 1) (dual of [11, 10, 2]-code) (see above)
- construction XX applied to C1 = C([1021,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(243, 1045, F2, 9) (dual of [1045, 1002, 10]-code), using
- appending kth column [i] based on linear OOA(243, 261, F2, 8, 9) (dual of [(261, 8), 2045, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(243, 261, F2, 9, 9) (dual of [(261, 9), 2306, 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 (34, 43, 261)-net over F2, using
- (u, u+v)-construction [i] based on
(252−19, 252, 1678095)-Net over F2 — Digital
Digital (233, 252, 1678095)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2252, 1678095, F2, 5, 19) (dual of [(1678095, 5), 8390223, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(244, 375, F2, 5, 9) (dual of [(375, 5), 1831, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(244, 375, F2, 2, 9) (dual of [(375, 2), 706, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(244, 523, F2, 2, 9) (dual of [(523, 2), 1002, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(244, 1046, F2, 9) (dual of [1046, 1002, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(244, 1047, F2, 9) (dual of [1047, 1003, 10]-code), using
- adding a parity check bit [i] based on linear OA(243, 1046, F2, 8) (dual of [1046, 1003, 9]-code), using
- construction XX applied to C1 = C([1021,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- linear OA(231, 1023, F2, 7) (dual of [1023, 992, 8]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(230, 1023, F2, 6) (dual of [1023, 993, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(241, 1023, F2, 9) (dual of [1023, 982, 10]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(220, 1023, F2, 4) (dual of [1023, 1003, 5]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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, 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 (see above)
- construction XX applied to C1 = C([1021,4]), C2 = C([1,6]), C3 = C1 + C2 = C([1,4]), and C∩ = C1 ∩ C2 = C([1021,6]) [i] based on
- adding a parity check bit [i] based on linear OA(243, 1046, F2, 8) (dual of [1046, 1003, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(244, 1047, F2, 9) (dual of [1047, 1003, 10]-code), using
- OOA 2-folding [i] based on linear OA(244, 1046, F2, 9) (dual of [1046, 1002, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(244, 523, F2, 2, 9) (dual of [(523, 2), 1002, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(244, 375, F2, 2, 9) (dual of [(375, 2), 706, 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(244, 375, F2, 5, 9) (dual of [(375, 5), 1831, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(252−19, 252, large)-Net in Base 2 — Upper bound on s
There is no (233, 252, large)-net in base 2, because
- 17 times m-reduction [i] would yield (233, 235, large)-net in base 2, but