Best Known (169−24, 169, s)-Nets in Base 3
(169−24, 169, 4924)-Net over F3 — Constructive and digital
Digital (145, 169, 4924)-net over F3, using
- net defined by OOA [i] based on linear OOA(3169, 4924, F3, 24, 24) (dual of [(4924, 24), 118007, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3169, 59088, F3, 24) (dual of [59088, 58919, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3169, 59091, F3, 24) (dual of [59091, 58922, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3161, 59050, F3, 25) (dual of [59050, 58889, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3169, 59091, F3, 24) (dual of [59091, 58922, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3169, 59088, F3, 24) (dual of [59088, 58919, 25]-code), using
(169−24, 169, 27002)-Net over F3 — Digital
Digital (145, 169, 27002)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3169, 27002, F3, 2, 24) (dual of [(27002, 2), 53835, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3169, 29545, F3, 2, 24) (dual of [(29545, 2), 58921, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3169, 59090, F3, 24) (dual of [59090, 58921, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3169, 59091, F3, 24) (dual of [59091, 58922, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3161, 59050, F3, 25) (dual of [59050, 58889, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3169, 59091, F3, 24) (dual of [59091, 58922, 25]-code), using
- OOA 2-folding [i] based on linear OA(3169, 59090, F3, 24) (dual of [59090, 58921, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(3169, 29545, F3, 2, 24) (dual of [(29545, 2), 58921, 25]-NRT-code), using
(169−24, 169, large)-Net in Base 3 — Upper bound on s
There is no (145, 169, large)-net in base 3, because
- 22 times m-reduction [i] would yield (145, 147, large)-net in base 3, but