Best Known (78−12, 78, s)-Nets in Base 128
(78−12, 78, 3844906)-Net over F128 — Constructive and digital
Digital (66, 78, 3844906)-net over F128, using
- generalized (u, u+v)-construction [i] based on
- digital (8, 12, 1048706)-net over F128, using
- net defined by OOA [i] based on linear OOA(12812, 1048706, F128, 4, 4) (dual of [(1048706, 4), 4194812, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(12812, 1048706, F128, 3, 4) (dual of [(1048706, 3), 3146106, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(1282, 129, F128, 3, 2) (dual of [(129, 3), 385, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;385,128) [i]
- linear OOA(12810, 1048577, F128, 3, 4) (dual of [(1048577, 3), 3145721, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(12810, 2097154, F128, 4) (dual of [2097154, 2097144, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- 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(1287, 2097152, F128, 3) (dual of [2097152, 2097145, 4]-code or 2097152-cap in PG(6,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(12810, 2097154, F128, 4) (dual of [2097154, 2097144, 5]-code), using
- linear OOA(1282, 129, F128, 3, 2) (dual of [(129, 3), 385, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(12812, 1048706, F128, 3, 4) (dual of [(1048706, 3), 3146106, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(12812, 1048706, F128, 4, 4) (dual of [(1048706, 4), 4194812, 5]-NRT-code), using
- digital (15, 21, 1398100)-net over F128, using
- s-reduction based on digital (15, 21, 2796201)-net over F128, using
- net defined by OOA [i] based on linear OOA(12821, 2796201, F128, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(12821, large, F128, 6) (dual of [large, large−21, 7]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 17895697 | 1284−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(12821, large, F128, 6) (dual of [large, large−21, 7]-code), using
- net defined by OOA [i] based on linear OOA(12821, 2796201, F128, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
- s-reduction based on digital (15, 21, 2796201)-net over F128, 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 (8, 12, 1048706)-net over F128, using
(78−12, 78, 5592400)-Net in Base 128 — Constructive
(66, 78, 5592400)-net in base 128, using
- 1281 times duplication [i] based on (65, 77, 5592400)-net in base 128, using
- (u, u+v)-construction [i] based on
- (13, 19, 2796201)-net in base 128, using
- net defined by OOA [i] based on OOA(12819, 2796201, S128, 6, 6), using
- OA 3-folding and stacking [i] 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
- OA 3-folding and stacking [i] based on OA(12819, large, S128, 6), using
- net defined by OOA [i] based on OOA(12819, 2796201, S128, 6, 6), using
- (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) (see above)
- 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
- net defined by OOA [i] based on OOA(12858, 2796200, S128, 14, 12), using
- (13, 19, 2796201)-net in base 128, using
- (u, u+v)-construction [i] based on
(78−12, 78, large)-Net over F128 — Digital
Digital (66, 78, large)-net over F128, using
- 1281 times duplication [i] based on digital (65, 77, large)-net over F128, using
- t-expansion [i] based on digital (56, 77, large)-net over F128, using
(78−12, 78, large)-Net in Base 128 — Upper bound on s
There is no (66, 78, large)-net in base 128, because
- 10 times m-reduction [i] would yield (66, 68, large)-net in base 128, but