Best Known (68−16, 68, s)-Nets in Base 128
(68−16, 68, 1048575)-Net over F128 — Constructive and digital
Digital (52, 68, 1048575)-net over F128, using
- 1283 times duplication [i] based on digital (49, 65, 1048575)-net over F128, using
- t-expansion [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
- t-expansion [i] based on digital (48, 65, 1048575)-net over F128, using
(68−16, 68, 1052671)-Net in Base 128 — Constructive
(52, 68, 1052671)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (7, 15, 4096)-net over F128, using
- net defined by OOA [i] based on linear OOA(12815, 4096, F128, 8, 8) (dual of [(4096, 8), 32753, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OA 4-folding and stacking [i] based on linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using
- net defined by OOA [i] based on linear OOA(12815, 4096, F128, 8, 8) (dual of [(4096, 8), 32753, 9]-NRT-code), using
- (37, 53, 1048575)-net in base 128, using
- net defined by OOA [i] based on OOA(12853, 1048575, S128, 16, 16), using
- OA 8-folding and stacking [i] based on OA(12853, 8388600, S128, 16), using
- discarding factors based on OA(12853, large, S128, 16), using
- discarding parts of the base [i] based on linear OA(25646, large, F256, 16) (dual of [large, large−46, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding parts of the base [i] based on linear OA(25646, large, F256, 16) (dual of [large, large−46, 17]-code), using
- discarding factors based on OA(12853, large, S128, 16), using
- OA 8-folding and stacking [i] based on OA(12853, 8388600, S128, 16), using
- net defined by OOA [i] based on OOA(12853, 1048575, S128, 16, 16), using
- digital (7, 15, 4096)-net over F128, using
(68−16, 68, large)-Net over F128 — Digital
Digital (52, 68, large)-net over F128, using
- t-expansion [i] based on digital (51, 68, large)-net over F128, using
- 2 times m-reduction [i] based on digital (51, 70, large)-net over F128, using
(68−16, 68, large)-Net in Base 128 — Upper bound on s
There is no (52, 68, large)-net in base 128, because
- 14 times m-reduction [i] would yield (52, 54, large)-net in base 128, but