Best Known (123−14, 123, s)-Nets in Base 3
(123−14, 123, 227764)-Net over F3 — Constructive and digital
Digital (109, 123, 227764)-net over F3, using
- 31 times duplication [i] based on digital (108, 122, 227764)-net over F3, using
- net defined by OOA [i] based on linear OOA(3122, 227764, F3, 14, 14) (dual of [(227764, 14), 3188574, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(3122, 1594348, F3, 14) (dual of [1594348, 1594226, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3122, 1594353, F3, 14) (dual of [1594353, 1594231, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(3118, 1594323, F3, 14) (dual of [1594323, 1594205, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(392, 1594323, F3, 11) (dual of [1594323, 1594231, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(34, 30, F3, 2) (dual of [30, 26, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(3122, 1594353, F3, 14) (dual of [1594353, 1594231, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(3122, 1594348, F3, 14) (dual of [1594348, 1594226, 15]-code), using
- net defined by OOA [i] based on linear OOA(3122, 227764, F3, 14, 14) (dual of [(227764, 14), 3188574, 15]-NRT-code), using
(123−14, 123, 531451)-Net over F3 — Digital
Digital (109, 123, 531451)-net over F3, using
- 31 times duplication [i] based on digital (108, 122, 531451)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3122, 531451, F3, 3, 14) (dual of [(531451, 3), 1594231, 15]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3122, 1594353, F3, 14) (dual of [1594353, 1594231, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(3118, 1594323, F3, 14) (dual of [1594323, 1594205, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(392, 1594323, F3, 11) (dual of [1594323, 1594231, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(34, 30, F3, 2) (dual of [30, 26, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- OOA 3-folding [i] based on linear OA(3122, 1594353, F3, 14) (dual of [1594353, 1594231, 15]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3122, 531451, F3, 3, 14) (dual of [(531451, 3), 1594231, 15]-NRT-code), using
(123−14, 123, large)-Net in Base 3 — Upper bound on s
There is no (109, 123, large)-net in base 3, because
- 12 times m-reduction [i] would yield (109, 111, large)-net in base 3, but