Best Known (49−11, 49, s)-Nets in Base 128
(49−11, 49, 1685848)-Net over F128 — Constructive and digital
Digital (38, 49, 1685848)-net over F128, using
- 1281 times duplication [i] based on digital (37, 48, 1685848)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 8128)-net over F128, using
- net defined by OOA [i] based on linear OOA(1287, 8128, F128, 5, 5) (dual of [(8128, 5), 40633, 6]-NRT-code), using
- appending kth column [i] based on 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
- appending kth column [i] based on linear OOA(1287, 8128, F128, 4, 5) (dual of [(8128, 4), 32505, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1287, 8128, F128, 5, 5) (dual of [(8128, 5), 40633, 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 (2, 7, 8128)-net over F128, using
- (u, u+v)-construction [i] based on
(49−11, 49, 2726297)-Net in Base 128 — Constructive
(38, 49, 2726297)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (8, 13, 1048577)-net over F128, using
- net defined by OOA [i] based on linear OOA(12813, 1048577, F128, 5, 5) (dual of [(1048577, 5), 5242872, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(12813, 1048577, F128, 4, 5) (dual of [(1048577, 4), 4194295, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(12813, 2097155, F128, 5) (dual of [2097155, 2097142, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(12813, 2097152, F128, 5) (dual of [2097152, 2097139, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(12813, 2097155, F128, 5) (dual of [2097155, 2097142, 6]-code), using
- appending kth column [i] based on linear OOA(12813, 1048577, F128, 4, 5) (dual of [(1048577, 4), 4194295, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(12813, 1048577, F128, 5, 5) (dual of [(1048577, 5), 5242872, 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 (8, 13, 1048577)-net over F128, using
(49−11, 49, large)-Net over F128 — Digital
Digital (38, 49, large)-net over F128, using
- 3 times m-reduction [i] based on digital (38, 52, large)-net over F128, using
(49−11, 49, large)-Net in Base 128 — Upper bound on s
There is no (38, 49, large)-net in base 128, because
- 9 times m-reduction [i] would yield (38, 40, large)-net in base 128, but