Best Known (52, 65, s)-Nets in Base 128
(52, 65, 2097151)-Net over F128 — Constructive and digital
Digital (52, 65, 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 (36, 49, 1398100)-net over F128, using
- net defined by OOA [i] based on linear OOA(12849, 1398100, F128, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12849, 8388601, F128, 13) (dual of [8388601, 8388552, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12849, large, F128, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(12849, large, F128, 13) (dual of [large, large−49, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12849, 8388601, F128, 13) (dual of [8388601, 8388552, 14]-code), using
- net defined by OOA [i] based on linear OOA(12849, 1398100, F128, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- digital (10, 16, 699051)-net over F128, using
(52, 65, 2796201)-Net in Base 128 — Constructive
(52, 65, 2796201)-net in base 128, using
- 1281 times duplication [i] based on (51, 64, 2796201)-net in base 128, using
- base change [i] based on digital (43, 56, 2796201)-net over F256, using
- net defined by OOA [i] based on linear OOA(25656, 2796201, F256, 15, 13) (dual of [(2796201, 15), 41942959, 14]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(25656, 5592403, F256, 3, 13) (dual of [(5592403, 3), 16777153, 14]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(25653, 5592402, F256, 3, 13) (dual of [(5592402, 3), 16777153, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- 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(25637, 2796201, F256, 3, 13) (dual of [(2796201, 3), 8388566, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- OOA 3-folding [i] based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- linear OOA(25616, 2796201, F256, 3, 6) (dual of [(2796201, 3), 8388587, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- 1 times NRT-code embedding in larger space [i] based on linear OOA(25653, 5592402, F256, 3, 13) (dual of [(5592402, 3), 16777153, 14]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(25656, 5592403, F256, 3, 13) (dual of [(5592403, 3), 16777153, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(25656, 2796201, F256, 15, 13) (dual of [(2796201, 15), 41942959, 14]-NRT-code), using
- base change [i] based on digital (43, 56, 2796201)-net over F256, using
(52, 65, large)-Net over F128 — Digital
Digital (52, 65, large)-net over F128, using
- t-expansion [i] based on digital (51, 65, large)-net over F128, using
- 5 times m-reduction [i] based on digital (51, 70, large)-net over F128, using
(52, 65, large)-Net in Base 128 — Upper bound on s
There is no (52, 65, large)-net in base 128, because
- 11 times m-reduction [i] would yield (52, 54, large)-net in base 128, but