Best Known (172−19, 172, s)-Nets in Base 4
(172−19, 172, 932067)-Net over F4 — Constructive and digital
Digital (153, 172, 932067)-net over F4, using
- net defined by OOA [i] based on linear OOA(4172, 932067, F4, 21, 19) (dual of [(932067, 21), 19573235, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(4172, 2796202, F4, 3, 19) (dual of [(2796202, 3), 8388434, 20]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4169, 2796201, F4, 3, 19) (dual of [(2796201, 3), 8388434, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4169, large, F4, 19) (dual of [large, large−169, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, 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(4169, large, F4, 19) (dual of [large, large−169, 20]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4169, 2796201, F4, 3, 19) (dual of [(2796201, 3), 8388434, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(4172, 2796202, F4, 3, 19) (dual of [(2796202, 3), 8388434, 20]-NRT-code), using
(172−19, 172, 4194302)-Net over F4 — Digital
Digital (153, 172, 4194302)-net over F4, using
- 41 times duplication [i] based on digital (152, 171, 4194302)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4171, 4194302, F4, 2, 19) (dual of [(4194302, 2), 8388433, 20]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4169, 4194301, F4, 2, 19) (dual of [(4194301, 2), 8388433, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4169, 8388602, F4, 19) (dual of [8388602, 8388433, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4169, large, F4, 19) (dual of [large, large−169, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(4169, large, F4, 19) (dual of [large, large−169, 20]-code), using
- OOA 2-folding [i] based on linear OA(4169, 8388602, F4, 19) (dual of [8388602, 8388433, 20]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4169, 4194301, F4, 2, 19) (dual of [(4194301, 2), 8388433, 20]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4171, 4194302, F4, 2, 19) (dual of [(4194302, 2), 8388433, 20]-NRT-code), using
(172−19, 172, large)-Net in Base 4 — Upper bound on s
There is no (153, 172, large)-net in base 4, because
- 17 times m-reduction [i] would yield (153, 155, large)-net in base 4, but