Best Known (49, 63, s)-Nets in Base 128
(49, 63, 1198629)-Net over F128 — Constructive and digital
Digital (49, 63, 1198629)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 258)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- digital (0, 7, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 3, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (39, 53, 1198371)-net over F128, using
- net defined by OOA [i] based on linear OOA(12853, 1198371, F128, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(12853, 8388597, F128, 14) (dual of [8388597, 8388544, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(12853, large, F128, 14) (dual of [large, large−53, 15]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9256395 | 1284−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(12853, large, F128, 14) (dual of [large, large−53, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(12853, 8388597, F128, 14) (dual of [8388597, 8388544, 15]-code), using
- net defined by OOA [i] based on linear OOA(12853, 1198371, F128, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- digital (3, 10, 258)-net over F128, using
(49, 63, 1220218)-Net in Base 128 — Constructive
(49, 63, 1220218)-net in base 128, using
- 1281 times duplication [i] based on (48, 62, 1220218)-net in base 128, using
- (u, u+v)-construction [i] based on
- (9, 16, 21847)-net in base 128, using
- base change [i] based on digital (7, 14, 21847)-net over F256, using
- net defined by OOA [i] based on linear OOA(25614, 21847, F256, 7, 7) (dual of [(21847, 7), 152915, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(25614, 21847, F256, 6, 7) (dual of [(21847, 6), 131068, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25614, 65542, F256, 7) (dual of [65542, 65528, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(25613, 65537, F256, 7) (dual of [65537, 65524, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(2569, 65537, F256, 5) (dual of [65537, 65528, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(25614, 65542, F256, 7) (dual of [65542, 65528, 8]-code), using
- appending kth column [i] based on linear OOA(25614, 21847, F256, 6, 7) (dual of [(21847, 6), 131068, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(25614, 21847, F256, 7, 7) (dual of [(21847, 7), 152915, 8]-NRT-code), using
- base change [i] based on digital (7, 14, 21847)-net over F256, using
- (32, 46, 1198371)-net in base 128, using
- net defined by OOA [i] based on OOA(12846, 1198371, S128, 14, 14), using
- OA 7-folding and stacking [i] based on OA(12846, 8388597, S128, 14), using
- discarding factors based on OA(12846, large, S128, 14), using
- discarding parts of the base [i] based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding parts of the base [i] based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- discarding factors based on OA(12846, large, S128, 14), using
- OA 7-folding and stacking [i] based on OA(12846, 8388597, S128, 14), using
- net defined by OOA [i] based on OOA(12846, 1198371, S128, 14, 14), using
- (9, 16, 21847)-net in base 128, using
- (u, u+v)-construction [i] based on
(49, 63, large)-Net over F128 — Digital
Digital (49, 63, large)-net over F128, using
- t-expansion [i] based on digital (48, 63, large)-net over F128, using
- 3 times m-reduction [i] based on digital (48, 66, large)-net over F128, using
(49, 63, large)-Net in Base 128 — Upper bound on s
There is no (49, 63, large)-net in base 128, because
- 12 times m-reduction [i] would yield (49, 51, large)-net in base 128, but