Best Known (63, 63+13, s)-Nets in Base 128
(63, 63+13, 2804328)-Net over F128 — Constructive and digital
Digital (63, 76, 2804328)-net over F128, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 6, 8128)-net over F128, using
- net defined by OOA [i] based on linear OOA(1286, 8128, F128, 4, 4) (dual of [(8128, 4), 32506, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1286, 16256, F128, 4) (dual of [16256, 16250, 5]-code), using
- 1 times truncation [i] based on linear OA(1287, 16257, F128, 5) (dual of [16257, 16250, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(1286, 16256, F128, 4) (dual of [16256, 16250, 5]-code), using
- net defined by OOA [i] based on linear OOA(1286, 8128, F128, 4, 4) (dual of [(8128, 4), 32506, 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 (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 (2, 6, 8128)-net over F128, using
(63, 63+13, 4194301)-Net in Base 128 — Constructive
(63, 76, 4194301)-net in base 128, using
- 1284 times duplication [i] based on (59, 72, 4194301)-net in base 128, using
- base change [i] based on digital (50, 63, 4194301)-net over F256, using
- net defined by OOA [i] based on linear OOA(25663, 4194301, F256, 15, 13) (dual of [(4194301, 15), 62914452, 14]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(25663, large, F256, 3, 13), 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(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(25610, 2796201, F256, 3, 4) (dual of [(2796201, 3), 8388593, 5]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OOA(25663, large, F256, 3, 13), using
- net defined by OOA [i] based on linear OOA(25663, 4194301, F256, 15, 13) (dual of [(4194301, 15), 62914452, 14]-NRT-code), using
- base change [i] based on digital (50, 63, 4194301)-net over F256, using
(63, 63+13, large)-Net over F128 — Digital
Digital (63, 76, large)-net over F128, using
- t-expansion [i] based on digital (56, 76, large)-net over F128, using
- 1 times m-reduction [i] based on digital (56, 77, large)-net over F128, using
(63, 63+13, large)-Net in Base 128 — Upper bound on s
There is no (63, 76, large)-net in base 128, because
- 11 times m-reduction [i] would yield (63, 65, large)-net in base 128, but