Best Known (56, 56+15, s)-Nets in Base 128
(56, 56+15, 1203834)-Net over F128 — Constructive and digital
Digital (56, 71, 1203834)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (7, 14, 5463)-net over F128, using
- net defined by OOA [i] based on linear OOA(12814, 5463, F128, 7, 7) (dual of [(5463, 7), 38227, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12814, 16390, F128, 7) (dual of [16390, 16376, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(12813, 16385, F128, 7) (dual of [16385, 16372, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(1289, 16385, F128, 5) (dual of [16385, 16376, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(12814, 16390, F128, 7) (dual of [16390, 16376, 8]-code), using
- net defined by OOA [i] based on linear OOA(12814, 5463, F128, 7, 7) (dual of [(5463, 7), 38227, 8]-NRT-code), using
- digital (42, 57, 1198371)-net over F128, using
- net defined by OOA [i] based on linear OOA(12857, 1198371, F128, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12857, 8388598, F128, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12857, large, F128, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(12857, large, F128, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12857, 8388598, F128, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(12857, 1198371, F128, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- digital (7, 14, 5463)-net over F128, using
(56, 56+15, 2097150)-Net in Base 128 — Constructive
(56, 71, 2097150)-net in base 128, using
- net defined by OOA [i] based on OOA(12871, 2097150, S128, 18, 15), using
- OOA 4-folding and stacking with additional row [i] based on OOA(12871, 8388601, S128, 2, 15), using
- discarding factors based on OOA(12871, 8388602, S128, 2, 15), using
- discarding parts of the base [i] based on linear OOA(25662, 8388602, F256, 2, 15) (dual of [(8388602, 2), 16777142, 16]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(25619, 4194301, F256, 2, 7) (dual of [(4194301, 2), 8388583, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25619, 8388602, F256, 7) (dual of [8388602, 8388583, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- OOA 2-folding [i] based on linear OA(25619, 8388602, F256, 7) (dual of [8388602, 8388583, 8]-code), using
- linear OOA(25643, 4194301, F256, 2, 15) (dual of [(4194301, 2), 8388559, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25643, 8388602, F256, 15) (dual of [8388602, 8388559, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 2-folding [i] based on linear OA(25643, 8388602, F256, 15) (dual of [8388602, 8388559, 16]-code), using
- linear OOA(25619, 4194301, F256, 2, 7) (dual of [(4194301, 2), 8388583, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding parts of the base [i] based on linear OOA(25662, 8388602, F256, 2, 15) (dual of [(8388602, 2), 16777142, 16]-NRT-code), using
- discarding factors based on OOA(12871, 8388602, S128, 2, 15), using
- OOA 4-folding and stacking with additional row [i] based on OOA(12871, 8388601, S128, 2, 15), using
(56, 56+15, large)-Net over F128 — Digital
Digital (56, 71, large)-net over F128, using
- 6 times m-reduction [i] based on digital (56, 77, large)-net over F128, using
(56, 56+15, large)-Net in Base 128 — Upper bound on s
There is no (56, 71, large)-net in base 128, because
- 13 times m-reduction [i] would yield (56, 58, large)-net in base 128, but