Best Known (55, 70, s)-Nets in Base 128
(55, 70, 1203832)-Net over F128 — Constructive and digital
Digital (55, 70, 1203832)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (6, 13, 5461)-net over F128, using
- net defined by OOA [i] based on linear OOA(12813, 5461, F128, 7, 7) (dual of [(5461, 7), 38214, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using
- net defined by OOA [i] based on linear OOA(12813, 5461, F128, 7, 7) (dual of [(5461, 7), 38214, 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 (6, 13, 5461)-net over F128, using
(55, 70, 1897424)-Net in Base 128 — Constructive
(55, 70, 1897424)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (13, 20, 699053)-net over F128, using
- net defined by OOA [i] based on linear OOA(12820, 699053, F128, 7, 7) (dual of [(699053, 7), 4893351, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12820, 2097160, F128, 7) (dual of [2097160, 2097140, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(12819, 2097153, F128, 7) (dual of [2097153, 2097134, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- 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(1281, 7, F128, 1) (dual of [7, 6, 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(12820, 2097160, F128, 7) (dual of [2097160, 2097140, 8]-code), using
- net defined by OOA [i] based on linear OOA(12820, 699053, F128, 7, 7) (dual of [(699053, 7), 4893351, 8]-NRT-code), using
- (35, 50, 1198371)-net in base 128, using
- net defined by OOA [i] based on OOA(12850, 1198371, S128, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12850, 8388598, S128, 15), using
- discarding factors based on OA(12850, large, S128, 15), using
- discarding parts of the base [i] 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 parts of the base [i] based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- discarding factors based on OA(12850, large, S128, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12850, 8388598, S128, 15), using
- net defined by OOA [i] based on OOA(12850, 1198371, S128, 15, 15), using
- digital (13, 20, 699053)-net over F128, using
(55, 70, large)-Net over F128 — Digital
Digital (55, 70, large)-net over F128, using
- t-expansion [i] based on digital (54, 70, large)-net over F128, using
- 4 times m-reduction [i] based on digital (54, 74, large)-net over F128, using
(55, 70, large)-Net in Base 128 — Upper bound on s
There is no (55, 70, large)-net in base 128, because
- 13 times m-reduction [i] would yield (55, 57, large)-net in base 128, but