Best Known (50−11, 50, s)-Nets in Base 128
(50−11, 50, 1685977)-Net over F128 — Constructive and digital
Digital (39, 50, 1685977)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (4, 9, 8257)-net over F128, using
- net defined by OOA [i] based on linear OOA(1289, 8257, F128, 5, 5) (dual of [(8257, 5), 41276, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(1289, 8257, F128, 4, 5) (dual of [(8257, 4), 33019, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(1282, 129, F128, 4, 2) (dual of [(129, 4), 514, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;514,128) [i]
- linear OOA(1287, 8128, F128, 4, 5) (dual of [(8128, 4), 32505, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(1287, 16257, F128, 5) (dual of [16257, 16250, 6]-code), using
- linear OOA(1282, 129, F128, 4, 2) (dual of [(129, 4), 514, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(1289, 8257, F128, 4, 5) (dual of [(8257, 4), 33019, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1289, 8257, F128, 5, 5) (dual of [(8257, 5), 41276, 6]-NRT-code), using
- digital (30, 41, 1677720)-net over F128, using
- net defined by OOA [i] based on linear OOA(12841, 1677720, F128, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(12841, 8388601, F128, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(12841, large, F128, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(12841, large, F128, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(12841, 8388601, F128, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(12841, 1677720, F128, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- digital (4, 9, 8257)-net over F128, using
(50−11, 50, 2726300)-Net in Base 128 — Constructive
(39, 50, 2726300)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (9, 14, 1048580)-net over F128, using
- net defined by OOA [i] based on linear OOA(12814, 1048580, F128, 5, 5) (dual of [(1048580, 5), 5242886, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(12814, 1048580, F128, 4, 5) (dual of [(1048580, 4), 4194306, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(12814, 2097161, F128, 5) (dual of [2097161, 2097147, 6]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(12813, 2097153, F128, 5) (dual of [2097153, 2097140, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(1287, 2097153, F128, 3) (dual of [2097153, 2097146, 4]-code or 2097153-cap in PG(6,128)), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(1287, 8, F128, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,128)), using
- dual of repetition code with length 8 [i]
- linear OA(1281, 8, F128, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- Reed–Solomon code RS(127,128) [i]
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(12814, 2097161, F128, 5) (dual of [2097161, 2097147, 6]-code), using
- appending kth column [i] based on linear OOA(12814, 1048580, F128, 4, 5) (dual of [(1048580, 4), 4194306, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(12814, 1048580, F128, 5, 5) (dual of [(1048580, 5), 5242886, 6]-NRT-code), using
- (25, 36, 1677720)-net in base 128, using
- net defined by OOA [i] based on OOA(12836, 1677720, S128, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(12836, 8388601, S128, 11), using
- discarding factors based on OA(12836, large, S128, 11), using
- discarding parts of the base [i] based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding parts of the base [i] based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- discarding factors based on OA(12836, large, S128, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(12836, 8388601, S128, 11), using
- net defined by OOA [i] based on OOA(12836, 1677720, S128, 11, 11), using
- digital (9, 14, 1048580)-net over F128, using
(50−11, 50, large)-Net over F128 — Digital
Digital (39, 50, large)-net over F128, using
- t-expansion [i] based on digital (38, 50, large)-net over F128, using
- 2 times m-reduction [i] based on digital (38, 52, large)-net over F128, using
(50−11, 50, large)-Net in Base 128 — Upper bound on s
There is no (39, 50, large)-net in base 128, because
- 9 times m-reduction [i] would yield (39, 41, large)-net in base 128, but