Best Known (189−24, 189, s)-Nets in Base 4
(189−24, 189, 87385)-Net over F4 — Constructive and digital
Digital (165, 189, 87385)-net over F4, using
- 41 times duplication [i] based on digital (164, 188, 87385)-net over F4, using
- net defined by OOA [i] based on linear OOA(4188, 87385, F4, 24, 24) (dual of [(87385, 24), 2097052, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(4188, 1048620, F4, 24) (dual of [1048620, 1048432, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4181, 1048577, F4, 25) (dual of [1048577, 1048396, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4141, 1048577, F4, 19) (dual of [1048577, 1048436, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- OA 12-folding and stacking [i] based on linear OA(4188, 1048620, F4, 24) (dual of [1048620, 1048432, 25]-code), using
- net defined by OOA [i] based on linear OOA(4188, 87385, F4, 24, 24) (dual of [(87385, 24), 2097052, 25]-NRT-code), using
(189−24, 189, 524312)-Net over F4 — Digital
Digital (165, 189, 524312)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4189, 524312, F4, 2, 24) (dual of [(524312, 2), 1048435, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4189, 1048624, F4, 24) (dual of [1048624, 1048435, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4189, 1048625, F4, 24) (dual of [1048625, 1048436, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4181, 1048577, F4, 25) (dual of [1048577, 1048396, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4141, 1048577, F4, 19) (dual of [1048577, 1048436, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(48, 48, F4, 4) (dual of [48, 40, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(48, 85, F4, 4) (dual of [85, 77, 5]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4189, 1048625, F4, 24) (dual of [1048625, 1048436, 25]-code), using
- OOA 2-folding [i] based on linear OA(4189, 1048624, F4, 24) (dual of [1048624, 1048435, 25]-code), using
(189−24, 189, large)-Net in Base 4 — Upper bound on s
There is no (165, 189, large)-net in base 4, because
- 22 times m-reduction [i] would yield (165, 167, large)-net in base 4, but