Best Known (49, 61, s)-Nets in Base 128
(49, 61, 2097151)-Net over F128 — Constructive and digital
Digital (49, 61, 2097151)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (10, 16, 699051)-net over F128, using
- net defined by OOA [i] based on linear OOA(12816, 699051, F128, 6, 6) (dual of [(699051, 6), 4194290, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(12816, 2097153, F128, 6) (dual of [2097153, 2097137, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(12816, 2097155, F128, 6) (dual of [2097155, 2097139, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(12813, 2097152, F128, 5) (dual of [2097152, 2097139, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(12816, 2097155, F128, 6) (dual of [2097155, 2097139, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(12816, 2097153, F128, 6) (dual of [2097153, 2097137, 7]-code), using
- net defined by OOA [i] based on linear OOA(12816, 699051, F128, 6, 6) (dual of [(699051, 6), 4194290, 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 (10, 16, 699051)-net over F128, using
(49, 61, 2796201)-Net in Base 128 — Constructive
(49, 61, 2796201)-net in base 128, using
- net defined by OOA [i] based on OOA(12861, 2796201, S128, 15, 12), using
- OOA 2-folding and stacking with additional row [i] based on OOA(12861, 5592403, S128, 3, 12), using
- 1 times NRT-code embedding in larger space [i] based on OOA(12858, 5592402, S128, 3, 12), using
- (u, u+v)-construction [i] based on
- OOA(12819, 2796201, S128, 3, 6), using
- OOA 3-folding [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
- OOA 3-folding [i] based on OA(12819, large, S128, 6), using
- OOA(12839, 2796201, S128, 3, 12), using
- OOA 3-folding [i] 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
- OOA 3-folding [i] based on OA(12839, large, S128, 12), using
- OOA(12819, 2796201, S128, 3, 6), using
- (u, u+v)-construction [i] based on
- 1 times NRT-code embedding in larger space [i] based on OOA(12858, 5592402, S128, 3, 12), using
- OOA 2-folding and stacking with additional row [i] based on OOA(12861, 5592403, S128, 3, 12), using
(49, 61, large)-Net over F128 — Digital
Digital (49, 61, large)-net over F128, using
- t-expansion [i] based on digital (48, 61, large)-net over F128, using
- 5 times m-reduction [i] based on digital (48, 66, large)-net over F128, using
(49, 61, large)-Net in Base 128 — Upper bound on s
There is no (49, 61, large)-net in base 128, because
- 10 times m-reduction [i] would yield (49, 51, large)-net in base 128, but