Best Known (37, 37+9, s)-Nets in Base 128
(37, 37+9, 4194300)-Net over F128 — Constructive and digital
Digital (37, 46, 4194300)-net over F128, using
- net defined by OOA [i] based on linear OOA(12846, 4194300, F128, 10, 9) (dual of [(4194300, 10), 41942954, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(12846, 8388601, F128, 2, 9) (dual of [(8388601, 2), 16777156, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12846, 8388602, F128, 2, 9) (dual of [(8388602, 2), 16777158, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(12813, 4194301, F128, 2, 4) (dual of [(4194301, 2), 8388589, 5]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12813, 8388602, F128, 4) (dual of [8388602, 8388589, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(12813, large, F128, 4) (dual of [large, large−13, 5]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 17895697 | 1284−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(12813, large, F128, 4) (dual of [large, large−13, 5]-code), using
- OOA 2-folding [i] based on linear OA(12813, 8388602, F128, 4) (dual of [8388602, 8388589, 5]-code), using
- linear OOA(12833, 4194301, F128, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12833, 8388602, F128, 9) (dual of [8388602, 8388569, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(12833, large, F128, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(12833, large, F128, 9) (dual of [large, large−33, 10]-code), using
- OOA 2-folding [i] based on linear OA(12833, 8388602, F128, 9) (dual of [8388602, 8388569, 10]-code), using
- linear OOA(12813, 4194301, F128, 2, 4) (dual of [(4194301, 2), 8388589, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(12846, 8388602, F128, 2, 9) (dual of [(8388602, 2), 16777158, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(12846, 8388601, F128, 2, 9) (dual of [(8388601, 2), 16777156, 10]-NRT-code), using
(37, 37+9, 4325636)-Net in Base 128 — Constructive
(37, 46, 4325636)-net in base 128, using
- net defined by OOA [i] based on OOA(12846, 4325636, S128, 12, 9), using
- OOA stacking with additional row [i] based on OOA(12846, 4325637, S128, 4, 9), using
- discarding parts of the base [i] based on linear OOA(25640, 4325637, F256, 4, 9) (dual of [(4325637, 4), 17302508, 10]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(2565, 131337, F256, 4, 3) (dual of [(131337, 4), 525343, 4]-NRT-code), using
- appending 1 arbitrary column [i] based on linear OOA(2565, 131337, F256, 3, 3) (dual of [(131337, 3), 394006, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(2565, 131337, F256, 2, 3) (dual of [(131337, 2), 262669, 4]-NRT-code), using
- appending 1 arbitrary column [i] based on linear OOA(2565, 131337, F256, 3, 3) (dual of [(131337, 3), 394006, 4]-NRT-code), using
- linear OOA(25610, 2097150, F256, 4, 4) (dual of [(2097150, 4), 8388590, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25610, 4194301, F256, 4, 4) (dual of [(4194301, 4), 16777194, 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, 4, 4) (dual of [(4194301, 4), 16777194, 5]-NRT-code), using
- linear OOA(25625, 2097150, F256, 4, 9) (dual of [(2097150, 4), 8388575, 10]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25625, 8388600, F256, 9) (dual of [8388600, 8388575, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- OOA 4-folding [i] based on linear OA(25625, 8388600, F256, 9) (dual of [8388600, 8388575, 10]-code), using
- linear OOA(2565, 131337, F256, 4, 3) (dual of [(131337, 4), 525343, 4]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- discarding parts of the base [i] based on linear OOA(25640, 4325637, F256, 4, 9) (dual of [(4325637, 4), 17302508, 10]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(12846, 4325637, S128, 4, 9), using
(37, 37+9, large)-Net over F128 — Digital
Digital (37, 46, large)-net over F128, using
- t-expansion [i] based on digital (35, 46, large)-net over F128, using
- 2 times m-reduction [i] based on digital (35, 48, large)-net over F128, using
(37, 37+9, large)-Net in Base 128 — Upper bound on s
There is no (37, 46, large)-net in base 128, because
- 7 times m-reduction [i] would yield (37, 39, large)-net in base 128, but