Best Known (169, 169+24, s)-Nets in Base 3
(169, 169+24, 44287)-Net over F3 — Constructive and digital
Digital (169, 193, 44287)-net over F3, using
- net defined by OOA [i] based on linear OOA(3193, 44287, F3, 24, 24) (dual of [(44287, 24), 1062695, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3193, 531444, F3, 24) (dual of [531444, 531251, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3193, 531453, F3, 24) (dual of [531453, 531260, 25]-code), using
- 1 times truncation [i] based on linear OA(3194, 531454, F3, 25) (dual of [531454, 531260, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3193, 531441, F3, 25) (dual of [531441, 531248, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3181, 531441, F3, 23) (dual of [531441, 531260, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(3194, 531454, F3, 25) (dual of [531454, 531260, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3193, 531453, F3, 24) (dual of [531453, 531260, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3193, 531444, F3, 24) (dual of [531444, 531251, 25]-code), using
(169, 169+24, 141537)-Net over F3 — Digital
Digital (169, 193, 141537)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3193, 141537, F3, 3, 24) (dual of [(141537, 3), 424418, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3193, 177151, F3, 3, 24) (dual of [(177151, 3), 531260, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3193, 531453, F3, 24) (dual of [531453, 531260, 25]-code), using
- 1 times truncation [i] based on linear OA(3194, 531454, F3, 25) (dual of [531454, 531260, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3193, 531441, F3, 25) (dual of [531441, 531248, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3181, 531441, F3, 23) (dual of [531441, 531260, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(3194, 531454, F3, 25) (dual of [531454, 531260, 26]-code), using
- OOA 3-folding [i] based on linear OA(3193, 531453, F3, 24) (dual of [531453, 531260, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(3193, 177151, F3, 3, 24) (dual of [(177151, 3), 531260, 25]-NRT-code), using
(169, 169+24, large)-Net in Base 3 — Upper bound on s
There is no (169, 193, large)-net in base 3, because
- 22 times m-reduction [i] would yield (169, 171, large)-net in base 3, but