Best Known (65−16, 65, s)-Nets in Base 128
(65−16, 65, 1048575)-Net over F128 — Constructive and digital
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
(65−16, 65, 1048834)-Net in Base 128 — Constructive
(49, 65, 1048834)-net in base 128, using
- 1281 times duplication [i] based on (48, 64, 1048834)-net in base 128, using
- base change [i] based on digital (40, 56, 1048834)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- digital (30, 46, 1048575)-net over F256, using
- net defined by OOA [i] based on linear OOA(25646, 1048575, F256, 16, 16) (dual of [(1048575, 16), 16777154, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(25646, 8388600, F256, 16) (dual of [8388600, 8388554, 17]-code), using
- discarding factors / shortening the dual code 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 factors / shortening the dual code based on linear OA(25646, large, F256, 16) (dual of [large, large−46, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(25646, 8388600, F256, 16) (dual of [8388600, 8388554, 17]-code), using
- net defined by OOA [i] based on linear OOA(25646, 1048575, F256, 16, 16) (dual of [(1048575, 16), 16777154, 17]-NRT-code), using
- digital (2, 10, 259)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (40, 56, 1048834)-net over F256, using
(65−16, 65, large)-Net over F128 — Digital
Digital (49, 65, large)-net over F128, using
- t-expansion [i] based on digital (48, 65, large)-net over F128, using
- 1 times m-reduction [i] based on digital (48, 66, large)-net over F128, using
(65−16, 65, large)-Net in Base 128 — Upper bound on s
There is no (49, 65, large)-net in base 128, because
- 14 times m-reduction [i] would yield (49, 51, large)-net in base 128, but