Best Known (77−18, 77, s)-Nets in Base 128
(77−18, 77, 932067)-Net over F128 — Constructive and digital
Digital (59, 77, 932067)-net over F128, using
- 1281 times duplication [i] based on digital (58, 76, 932067)-net over F128, using
- t-expansion [i] based on digital (57, 76, 932067)-net over F128, using
- net defined by OOA [i] based on linear OOA(12876, 932067, F128, 21, 19) (dual of [(932067, 21), 19573331, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(12876, 2796202, F128, 3, 19) (dual of [(2796202, 3), 8388530, 20]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(12873, 2796201, F128, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12873, large, F128, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(12873, large, F128, 19) (dual of [large, large−73, 20]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(12873, 2796201, F128, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(12876, 2796202, F128, 3, 19) (dual of [(2796202, 3), 8388530, 20]-NRT-code), using
- net defined by OOA [i] based on linear OOA(12876, 932067, F128, 21, 19) (dual of [(932067, 21), 19573331, 20]-NRT-code), using
- t-expansion [i] based on digital (57, 76, 932067)-net over F128, using
(77−18, 77, 936163)-Net in Base 128 — Constructive
(59, 77, 936163)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (8, 17, 4096)-net over F128, using
- net defined by OOA [i] based on linear OOA(12817, 4096, F128, 9, 9) (dual of [(4096, 9), 36847, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12817, 16385, F128, 9) (dual of [16385, 16368, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(12817, 16385, F128, 9) (dual of [16385, 16368, 10]-code), using
- net defined by OOA [i] based on linear OOA(12817, 4096, F128, 9, 9) (dual of [(4096, 9), 36847, 10]-NRT-code), using
- (42, 60, 932067)-net in base 128, using
- net defined by OOA [i] based on OOA(12860, 932067, S128, 18, 18), using
- OA 9-folding and stacking [i] based on OA(12860, large, S128, 18), using
- discarding parts of the base [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding parts of the base [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- OA 9-folding and stacking [i] based on OA(12860, large, S128, 18), using
- net defined by OOA [i] based on OOA(12860, 932067, S128, 18, 18), using
- digital (8, 17, 4096)-net over F128, using
(77−18, 77, large)-Net over F128 — Digital
Digital (59, 77, large)-net over F128, using
- t-expansion [i] based on digital (56, 77, large)-net over F128, using
(77−18, 77, large)-Net in Base 128 — Upper bound on s
There is no (59, 77, large)-net in base 128, because
- 16 times m-reduction [i] would yield (59, 61, large)-net in base 128, but