Best Known (35, 35+14, s)-Nets in Base 128
(35, 35+14, 299744)-Net over F128 — Constructive and digital
Digital (35, 49, 299744)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 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 (27, 41, 299594)-net over F128, using
- net defined by OOA [i] based on linear OOA(12841, 299594, F128, 14, 14) (dual of [(299594, 14), 4194275, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(12841, 2097158, F128, 14) (dual of [2097158, 2097117, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(12841, 2097159, F128, 14) (dual of [2097159, 2097118, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(12841, 2097159, F128, 14) (dual of [2097159, 2097118, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(12841, 2097158, F128, 14) (dual of [2097158, 2097117, 15]-code), using
- net defined by OOA [i] based on linear OOA(12841, 299594, F128, 14, 14) (dual of [(299594, 14), 4194275, 15]-NRT-code), using
- digital (1, 8, 150)-net over F128, using
(35, 35+14, 1198371)-Net in Base 128 — Constructive
(35, 49, 1198371)-net in base 128, using
- 1 times m-reduction [i] based on (35, 50, 1198371)-net in base 128, using
- net defined by OOA [i] based on OOA(12850, 1198371, S128, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12850, 8388598, S128, 15), using
- discarding factors based on OA(12850, large, S128, 15), using
- discarding parts of the base [i] 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 parts of the base [i] based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- discarding factors based on OA(12850, large, S128, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12850, 8388598, S128, 15), using
- net defined by OOA [i] based on OOA(12850, 1198371, S128, 15, 15), using
(35, 35+14, 3909884)-Net over F128 — Digital
Digital (35, 49, 3909884)-net over F128, using
(35, 35+14, 4194302)-Net in Base 128
(35, 49, 4194302)-net in base 128, using
- 1281 times duplication [i] based on (34, 48, 4194302)-net in base 128, using
- base change [i] based on digital (28, 42, 4194302)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25642, 4194302, F256, 2, 14) (dual of [(4194302, 2), 8388562, 15]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(25640, 4194301, F256, 2, 14) (dual of [(4194301, 2), 8388562, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25640, 8388602, F256, 14) (dual of [8388602, 8388562, 15]-code), using
- discarding factors / shortening the dual code 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 factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- OOA 2-folding [i] based on linear OA(25640, 8388602, F256, 14) (dual of [8388602, 8388562, 15]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(25640, 4194301, F256, 2, 14) (dual of [(4194301, 2), 8388562, 15]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25642, 4194302, F256, 2, 14) (dual of [(4194302, 2), 8388562, 15]-NRT-code), using
- base change [i] based on digital (28, 42, 4194302)-net over F256, using
(35, 35+14, large)-Net in Base 128 — Upper bound on s
There is no (35, 49, large)-net in base 128, because
- 12 times m-reduction [i] would yield (35, 37, large)-net in base 128, but