Best Known (57, 57+12, s)-Nets in Base 128
(57, 57+12, 2796201)-Net over F128 — Constructive and digital
Digital (57, 69, 2796201)-net over F128, using
- net defined by OOA [i] based on linear OOA(12869, 2796201, F128, 15, 12) (dual of [(2796201, 15), 41942946, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(12869, 5592403, F128, 3, 12) (dual of [(5592403, 3), 16777140, 13]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(12866, 5592402, F128, 3, 12) (dual of [(5592402, 3), 16777140, 13]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(12821, 2796201, F128, 3, 6) (dual of [(2796201, 3), 8388582, 7]-NRT-code), using
- OOA 3-folding [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]
- OOA 3-folding [i] based on linear OA(12821, large, F128, 6) (dual of [large, large−21, 7]-code), using
- linear OOA(12845, 2796201, F128, 3, 12) (dual of [(2796201, 3), 8388558, 13]-NRT-code), using
- OOA 3-folding [i] 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]
- OOA 3-folding [i] based on linear OA(12845, large, F128, 12) (dual of [large, large−45, 13]-code), using
- linear OOA(12821, 2796201, F128, 3, 6) (dual of [(2796201, 3), 8388582, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- 1 times NRT-code embedding in larger space [i] based on linear OOA(12866, 5592402, F128, 3, 12) (dual of [(5592402, 3), 16777140, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(12869, 5592403, F128, 3, 12) (dual of [(5592403, 3), 16777140, 13]-NRT-code), using
(57, 57+12, 4194301)-Net in Base 128 — Constructive
(57, 69, 4194301)-net in base 128, using
- net defined by OOA [i] based on OOA(12869, 4194301, S128, 15, 12), using
- OOA 2-folding and stacking with additional row [i] based on OOA(12869, large, S128, 3, 12), using
- discarding parts of the base [i] based on linear OOA(25660, large, F256, 3, 12), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(25610, 2796201, F256, 3, 4) (dual of [(2796201, 3), 8388593, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25610, 4194301, F256, 3, 4) (dual of [(4194301, 3), 12582893, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(25610, 8388602, F256, 4) (dual of [8388602, 8388592, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(25610, large, F256, 4) (dual of [large, large−10, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(25610, large, F256, 4) (dual of [large, large−10, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(25610, 8388602, F256, 4) (dual of [8388602, 8388592, 5]-code), using
- discarding factors / shortening the dual code based on linear OOA(25610, 4194301, F256, 3, 4) (dual of [(4194301, 3), 12582893, 5]-NRT-code), using
- linear OOA(25616, 2796201, F256, 3, 6) (dual of [(2796201, 3), 8388587, 7]-NRT-code), using
- OOA 3-folding [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]
- OOA 3-folding [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- linear OOA(25634, 2796201, F256, 3, 12) (dual of [(2796201, 3), 8388569, 13]-NRT-code), using
- OOA 3-folding [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]
- OOA 3-folding [i] based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- linear OOA(25610, 2796201, F256, 3, 4) (dual of [(2796201, 3), 8388593, 5]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- discarding parts of the base [i] based on linear OOA(25660, large, F256, 3, 12), using
- OOA 2-folding and stacking with additional row [i] based on OOA(12869, large, S128, 3, 12), using
(57, 57+12, large)-Net over F128 — Digital
Digital (57, 69, large)-net over F128, using
- t-expansion [i] based on digital (56, 69, large)-net over F128, using
- 8 times m-reduction [i] based on digital (56, 77, large)-net over F128, using
(57, 57+12, large)-Net in Base 128 — Upper bound on s
There is no (57, 69, large)-net in base 128, because
- 10 times m-reduction [i] would yield (57, 59, large)-net in base 128, but