Best Known (41, 41+15, s)-Nets in Base 128
(41, 41+15, 305054)-Net over F128 — Constructive and digital
Digital (41, 56, 305054)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (6, 13, 5461)-net over F128, using
- net defined by OOA [i] based on linear OOA(12813, 5461, F128, 7, 7) (dual of [(5461, 7), 38214, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using
- net defined by OOA [i] based on linear OOA(12813, 5461, F128, 7, 7) (dual of [(5461, 7), 38214, 8]-NRT-code), using
- digital (28, 43, 299593)-net over F128, using
- net defined by OOA [i] based on linear OOA(12843, 299593, F128, 15, 15) (dual of [(299593, 15), 4493852, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using
- net defined by OOA [i] based on linear OOA(12843, 299593, F128, 15, 15) (dual of [(299593, 15), 4493852, 16]-NRT-code), using
- digital (6, 13, 5461)-net over F128, using
(41, 41+15, 1198371)-Net in Base 128 — Constructive
(41, 56, 1198371)-net in base 128, using
- base change [i] based on digital (34, 49, 1198371)-net over F256, using
- 2566 times duplication [i] based on digital (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- 2566 times duplication [i] based on digital (28, 43, 1198371)-net over F256, using
(41, 41+15, large)-Net over F128 — Digital
Digital (41, 56, large)-net over F128, using
- 1281 times duplication [i] based on digital (40, 55, large)-net over F128, using
(41, 41+15, large)-Net in Base 128 — Upper bound on s
There is no (41, 56, large)-net in base 128, because
- 13 times m-reduction [i] would yield (41, 43, large)-net in base 128, but