Best Known (53, 70, s)-Nets in Base 128
(53, 70, 1048575)-Net over F128 — Constructive and digital
Digital (53, 70, 1048575)-net over F128, using
- 1285 times duplication [i] based on digital (48, 65, 1048575)-net over F128, using
- net defined by OOA [i] based on linear OOA(12865, 1048575, F128, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12865, 8388601, F128, 17) (dual of [8388601, 8388536, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(12865, large, F128, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(12865, large, F128, 17) (dual of [large, large−65, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12865, 8388601, F128, 17) (dual of [8388601, 8388536, 18]-code), using
- net defined by OOA [i] based on linear OOA(12865, 1048575, F128, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
(53, 70, 1049088)-Net in Base 128 — Constructive
(53, 70, 1049088)-net in base 128, using
- (u, u+v)-construction [i] based on
- (6, 14, 513)-net in base 128, using
- net defined by OOA [i] based on OOA(12814, 513, S128, 12, 8), using
- OOA stacking with additional row [i] based on OOA(12814, 514, S128, 4, 8), using
- discarding parts of the base [i] based on linear OOA(25612, 514, F256, 4, 8) (dual of [(514, 4), 2044, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(2564, 257, F256, 4, 4) (dual of [(257, 4), 1024, 5]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;1024,256) [i]
- linear OOA(2568, 257, F256, 4, 8) (dual of [(257, 4), 1020, 9]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;1020,256) [i]
- linear OOA(2564, 257, F256, 4, 4) (dual of [(257, 4), 1024, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding parts of the base [i] based on linear OOA(25612, 514, F256, 4, 8) (dual of [(514, 4), 2044, 9]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(12814, 514, S128, 4, 8), using
- net defined by OOA [i] based on OOA(12814, 513, S128, 12, 8), using
- (39, 56, 1048575)-net in base 128, using
- base change [i] based on digital (32, 49, 1048575)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 1048575, F256, 17, 17) (dual of [(1048575, 17), 17825726, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25649, 8388601, F256, 17) (dual of [8388601, 8388552, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25649, 8388601, F256, 17) (dual of [8388601, 8388552, 18]-code), using
- net defined by OOA [i] based on linear OOA(25649, 1048575, F256, 17, 17) (dual of [(1048575, 17), 17825726, 18]-NRT-code), using
- base change [i] based on digital (32, 49, 1048575)-net over F256, using
- (6, 14, 513)-net in base 128, using
(53, 70, large)-Net over F128 — Digital
Digital (53, 70, large)-net over F128, using
- t-expansion [i] based on digital (51, 70, large)-net over F128, using
(53, 70, large)-Net in Base 128 — Upper bound on s
There is no (53, 70, large)-net in base 128, because
- 15 times m-reduction [i] would yield (53, 55, large)-net in base 128, but