Best Known (19, 27, s)-Nets in Base 128
(19, 27, 524438)-Net over F128 — Constructive and digital
Digital (19, 27, 524438)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- digital (14, 22, 524288)-net over F128, using
- net defined by OOA [i] based on linear OOA(12822, 524288, F128, 8, 8) (dual of [(524288, 8), 4194282, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OA 4-folding and stacking [i] based on linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using
- net defined by OOA [i] based on linear OOA(12822, 524288, F128, 8, 8) (dual of [(524288, 8), 4194282, 9]-NRT-code), using
- digital (1, 5, 150)-net over F128, using
(19, 27, 2097150)-Net in Base 128 — Constructive
(19, 27, 2097150)-net in base 128, using
- 1281 times duplication [i] based on (18, 26, 2097150)-net in base 128, using
- net defined by OOA [i] based on OOA(12826, 2097150, S128, 8, 8), using
- OA 4-folding and stacking [i] based on OA(12826, 8388600, S128, 8), using
- discarding factors based on OA(12826, large, S128, 8), using
- discarding parts of the base [i] based on linear OA(25622, large, F256, 8) (dual of [large, large−22, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding parts of the base [i] based on linear OA(25622, large, F256, 8) (dual of [large, large−22, 9]-code), using
- discarding factors based on OA(12826, large, S128, 8), using
- OA 4-folding and stacking [i] based on OA(12826, 8388600, S128, 8), using
- net defined by OOA [i] based on OOA(12826, 2097150, S128, 8, 8), using
(19, 27, 3572113)-Net over F128 — Digital
Digital (19, 27, 3572113)-net over F128, using
(19, 27, 7941361)-Net in Base 128
(19, 27, 7941361)-net in base 128, using
- net defined by OOA [i] based on OOA(12827, 7941361, S128, 12, 8), using
- OOA stacking with additional row [i] based on OOA(12827, 7941362, S128, 4, 8), using
- discarding parts of the base [i] based on linear OOA(25623, 7941362, F256, 4, 8) (dual of [(7941362, 4), 31765425, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25623, 7941362, F256, 8) (dual of [7941362, 7941339, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(25623, large, F256, 8) (dual of [large, large−23, 9]-code), using
- 1 times code embedding in larger space [i] based on linear OA(25622, large, F256, 8) (dual of [large, large−22, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 1 times code embedding in larger space [i] based on linear OA(25622, large, F256, 8) (dual of [large, large−22, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(25623, large, F256, 8) (dual of [large, large−23, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25623, 7941362, F256, 8) (dual of [7941362, 7941339, 9]-code), using
- discarding parts of the base [i] based on linear OOA(25623, 7941362, F256, 4, 8) (dual of [(7941362, 4), 31765425, 9]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(12827, 7941362, S128, 4, 8), using
(19, 27, large)-Net in Base 128 — Upper bound on s
There is no (19, 27, large)-net in base 128, because
- 6 times m-reduction [i] would yield (19, 21, large)-net in base 128, but