Best Known (162−20, 162, s)-Nets in Base 3
(162−20, 162, 53147)-Net over F3 — Constructive and digital
Digital (142, 162, 53147)-net over F3, using
- net defined by OOA [i] based on linear OOA(3162, 53147, F3, 20, 20) (dual of [(53147, 20), 1062778, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3162, 531470, F3, 20) (dual of [531470, 531308, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3161, 531469, F3, 20) (dual of [531469, 531308, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3133, 531441, F3, 17) (dual of [531441, 531308, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(34, 28, F3, 2) (dual of [28, 24, 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(19) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3161, 531469, F3, 20) (dual of [531469, 531308, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3162, 531470, F3, 20) (dual of [531470, 531308, 21]-code), using
(162−20, 162, 177156)-Net over F3 — Digital
Digital (142, 162, 177156)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3162, 177156, F3, 3, 20) (dual of [(177156, 3), 531306, 21]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3161, 177156, F3, 3, 20) (dual of [(177156, 3), 531307, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3161, 531468, F3, 20) (dual of [531468, 531307, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3161, 531469, F3, 20) (dual of [531469, 531308, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3133, 531441, F3, 17) (dual of [531441, 531308, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(34, 28, F3, 2) (dual of [28, 24, 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(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3161, 531469, F3, 20) (dual of [531469, 531308, 21]-code), using
- OOA 3-folding [i] based on linear OA(3161, 531468, F3, 20) (dual of [531468, 531307, 21]-code), using
- 31 times duplication [i] based on linear OOA(3161, 177156, F3, 3, 20) (dual of [(177156, 3), 531307, 21]-NRT-code), using
(162−20, 162, large)-Net in Base 3 — Upper bound on s
There is no (142, 162, large)-net in base 3, because
- 18 times m-reduction [i] would yield (142, 144, large)-net in base 3, but