Best Known (58−12, 58, s)-Nets in Base 128
(58−12, 58, 1403564)-Net over F128 — Constructive and digital
Digital (46, 58, 1403564)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (7, 13, 5464)-net over F128, using
- net defined by OOA [i] based on linear OOA(12813, 5464, F128, 6, 6) (dual of [(5464, 6), 32771, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(12813, 16392, F128, 6) (dual of [16392, 16379, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(12811, 16384, F128, 6) (dual of [16384, 16373, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1285, 16384, F128, 3) (dual of [16384, 16379, 4]-code or 16384-cap in PG(4,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- OA 3-folding and stacking [i] based on linear OA(12813, 16392, F128, 6) (dual of [16392, 16379, 7]-code), using
- net defined by OOA [i] based on linear OOA(12813, 5464, F128, 6, 6) (dual of [(5464, 6), 32771, 7]-NRT-code), using
- digital (33, 45, 1398100)-net over F128, using
- net defined by OOA [i] based on linear OOA(12845, 1398100, F128, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(12845, 8388600, F128, 12) (dual of [8388600, 8388555, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(12845, large, F128, 12) (dual of [large, large−45, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9256395 | 1284−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(12845, large, F128, 12) (dual of [large, large−45, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(12845, 8388600, F128, 12) (dual of [8388600, 8388555, 13]-code), using
- net defined by OOA [i] based on linear OOA(12845, 1398100, F128, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- digital (7, 13, 5464)-net over F128, using
(58−12, 58, 2796200)-Net in Base 128 — Constructive
(46, 58, 2796200)-net in base 128, using
- net defined by OOA [i] based on OOA(12858, 2796200, S128, 14, 12), using
- OOA 3-folding and stacking with additional row [i] based on OOA(12858, 8388601, S128, 2, 12), using
- discarding factors based on OOA(12858, 8388602, S128, 2, 12), using
- (u, u+v)-construction [i] based on
- OOA(12819, 4194301, S128, 2, 6), using
- OOA 2-folding [i] based on OA(12819, 8388602, S128, 6), using
- discarding factors based on OA(12819, large, S128, 6), using
- discarding parts of the base [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding parts of the base [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- discarding factors based on OA(12819, large, S128, 6), using
- OOA 2-folding [i] based on OA(12819, 8388602, S128, 6), using
- OOA(12839, 4194301, S128, 2, 12), using
- OOA 2-folding [i] based on OA(12839, 8388602, S128, 12), using
- discarding factors based on OA(12839, large, S128, 12), using
- discarding parts of the base [i] based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding parts of the base [i] based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- discarding factors based on OA(12839, large, S128, 12), using
- OOA 2-folding [i] based on OA(12839, 8388602, S128, 12), using
- OOA(12819, 4194301, S128, 2, 6), using
- (u, u+v)-construction [i] based on
- discarding factors based on OOA(12858, 8388602, S128, 2, 12), using
- OOA 3-folding and stacking with additional row [i] based on OOA(12858, 8388601, S128, 2, 12), using
(58−12, 58, large)-Net over F128 — Digital
Digital (46, 58, large)-net over F128, using
- 5 times m-reduction [i] based on digital (46, 63, large)-net over F128, using
(58−12, 58, large)-Net in Base 128 — Upper bound on s
There is no (46, 58, large)-net in base 128, because
- 10 times m-reduction [i] would yield (46, 48, large)-net in base 128, but