Best Known (78−16, 78, s)-Nets in Base 128
(78−16, 78, 1052673)-Net over F128 — Constructive and digital
Digital (62, 78, 1052673)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (9, 17, 4098)-net over F128, using
- net defined by OOA [i] based on linear OOA(12817, 4098, F128, 8, 8) (dual of [(4098, 8), 32767, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(12817, 16392, F128, 8) (dual of [16392, 16375, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1289, 16384, F128, 5) (dual of [16384, 16375, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(7) ⊂ Ce(4) [i] based on
- OA 4-folding and stacking [i] based on linear OA(12817, 16392, F128, 8) (dual of [16392, 16375, 9]-code), using
- net defined by OOA [i] based on linear OOA(12817, 4098, F128, 8, 8) (dual of [(4098, 8), 32767, 9]-NRT-code), using
- digital (45, 61, 1048575)-net over F128, using
- net defined by OOA [i] based on linear OOA(12861, 1048575, F128, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(12861, 8388600, F128, 16) (dual of [8388600, 8388539, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(12861, large, F128, 16) (dual of [large, large−61, 17]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9256395 | 1284−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(12861, large, F128, 16) (dual of [large, large−61, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(12861, 8388600, F128, 16) (dual of [8388600, 8388539, 17]-code), using
- net defined by OOA [i] based on linear OOA(12861, 1048575, F128, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- digital (9, 17, 4098)-net over F128, using
(78−16, 78, 2097150)-Net in Base 128 — Constructive
(62, 78, 2097150)-net in base 128, using
- net defined by OOA [i] based on OOA(12878, 2097150, S128, 18, 16), using
- OOA 4-folding and stacking with additional row [i] based on OOA(12878, 8388601, S128, 2, 16), using
- discarding factors based on OOA(12878, 8388602, S128, 2, 16), using
- discarding parts of the base [i] based on linear OOA(25668, 8388602, F256, 2, 16) (dual of [(8388602, 2), 16777136, 17]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(25622, 4194301, F256, 2, 8) (dual of [(4194301, 2), 8388580, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25622, 8388602, F256, 8) (dual of [8388602, 8388580, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(25622, large, F256, 8) (dual of [large, large−22, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(25622, large, F256, 8) (dual of [large, large−22, 9]-code), using
- OOA 2-folding [i] based on linear OA(25622, 8388602, F256, 8) (dual of [8388602, 8388580, 9]-code), using
- linear OOA(25646, 4194301, F256, 2, 16) (dual of [(4194301, 2), 8388556, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25646, 8388602, F256, 16) (dual of [8388602, 8388556, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(25646, large, F256, 16) (dual of [large, large−46, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(25646, large, F256, 16) (dual of [large, large−46, 17]-code), using
- OOA 2-folding [i] based on linear OA(25646, 8388602, F256, 16) (dual of [8388602, 8388556, 17]-code), using
- linear OOA(25622, 4194301, F256, 2, 8) (dual of [(4194301, 2), 8388580, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding parts of the base [i] based on linear OOA(25668, 8388602, F256, 2, 16) (dual of [(8388602, 2), 16777136, 17]-NRT-code), using
- discarding factors based on OOA(12878, 8388602, S128, 2, 16), using
- OOA 4-folding and stacking with additional row [i] based on OOA(12878, 8388601, S128, 2, 16), using
(78−16, 78, large)-Net over F128 — Digital
Digital (62, 78, large)-net over F128, using
- 1281 times duplication [i] based on digital (61, 77, large)-net over F128, using
- t-expansion [i] based on digital (56, 77, large)-net over F128, using
(78−16, 78, large)-Net in Base 128 — Upper bound on s
There is no (62, 78, large)-net in base 128, because
- 14 times m-reduction [i] would yield (62, 64, large)-net in base 128, but