Best Known (165, 165+21, s)-Nets in Base 3
(165, 165+21, 159435)-Net over F3 — Constructive and digital
Digital (165, 186, 159435)-net over F3, using
- 32 times duplication [i] based on digital (163, 184, 159435)-net over F3, using
- net defined by OOA [i] based on linear OOA(3184, 159435, F3, 21, 21) (dual of [(159435, 21), 3347951, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3184, 1594351, F3, 21) (dual of [1594351, 1594167, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3183, 1594324, F3, 21) (dual of [1594324, 1594141, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 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(31, 27, F3, 1) (dual of [27, 26, 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 C([0,10]) ⊂ C([0,9]) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(3184, 1594351, F3, 21) (dual of [1594351, 1594167, 22]-code), using
- net defined by OOA [i] based on linear OOA(3184, 159435, F3, 21, 21) (dual of [(159435, 21), 3347951, 22]-NRT-code), using
(165, 165+21, 490875)-Net over F3 — Digital
Digital (165, 186, 490875)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3186, 490875, F3, 3, 21) (dual of [(490875, 3), 1472439, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3186, 531451, F3, 3, 21) (dual of [(531451, 3), 1594167, 22]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3185, 531451, F3, 3, 21) (dual of [(531451, 3), 1594168, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3185, 1594353, F3, 21) (dual of [1594353, 1594168, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3184, 1594352, F3, 21) (dual of [1594352, 1594168, 22]-code), using
- construction X4 applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3183, 1594324, F3, 21) (dual of [1594324, 1594141, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 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(327, 28, F3, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,3)), using
- dual of repetition code with length 28 [i]
- linear OA(31, 28, F3, 1) (dual of [28, 27, 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 X4 applied to C([0,10]) ⊂ C([0,9]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3184, 1594352, F3, 21) (dual of [1594352, 1594168, 22]-code), using
- OOA 3-folding [i] based on linear OA(3185, 1594353, F3, 21) (dual of [1594353, 1594168, 22]-code), using
- 31 times duplication [i] based on linear OOA(3185, 531451, F3, 3, 21) (dual of [(531451, 3), 1594168, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3186, 531451, F3, 3, 21) (dual of [(531451, 3), 1594167, 22]-NRT-code), using
(165, 165+21, large)-Net in Base 3 — Upper bound on s
There is no (165, 186, large)-net in base 3, because
- 19 times m-reduction [i] would yield (165, 167, large)-net in base 3, but