Best Known (145, 145+19, s)-Nets in Base 3
(145, 145+19, 177150)-Net over F3 — Constructive and digital
Digital (145, 164, 177150)-net over F3, using
- 33 times duplication [i] based on digital (142, 161, 177150)-net over F3, using
- net defined by OOA [i] based on linear OOA(3161, 177150, F3, 19, 19) (dual of [(177150, 19), 3365689, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3161, 1594351, F3, 19) (dual of [1594351, 1594190, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3161, 1594353, F3, 19) (dual of [1594353, 1594192, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3131, 1594323, F3, 16) (dual of [1594323, 1594192, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3161, 1594353, F3, 19) (dual of [1594353, 1594192, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3161, 1594351, F3, 19) (dual of [1594351, 1594190, 20]-code), using
- net defined by OOA [i] based on linear OOA(3161, 177150, F3, 19, 19) (dual of [(177150, 19), 3365689, 20]-NRT-code), using
(145, 145+19, 424447)-Net over F3 — Digital
Digital (145, 164, 424447)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3164, 424447, F3, 3, 19) (dual of [(424447, 3), 1273177, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3164, 531452, F3, 3, 19) (dual of [(531452, 3), 1594192, 20]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3163, 531452, F3, 3, 19) (dual of [(531452, 3), 1594193, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3163, 1594356, F3, 19) (dual of [1594356, 1594193, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(3157, 1594324, F3, 19) (dual of [1594324, 1594167, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(3131, 1594324, F3, 15) (dual of [1594324, 1594193, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(36, 32, F3, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- OOA 3-folding [i] based on linear OA(3163, 1594356, F3, 19) (dual of [1594356, 1594193, 20]-code), using
- 31 times duplication [i] based on linear OOA(3163, 531452, F3, 3, 19) (dual of [(531452, 3), 1594193, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3164, 531452, F3, 3, 19) (dual of [(531452, 3), 1594192, 20]-NRT-code), using
(145, 145+19, large)-Net in Base 3 — Upper bound on s
There is no (145, 164, large)-net in base 3, because
- 17 times m-reduction [i] would yield (145, 147, large)-net in base 3, but