Best Known (147, 147+27, s)-Nets in Base 3
(147, 147+27, 1517)-Net over F3 — Constructive and digital
Digital (147, 174, 1517)-net over F3, using
- 33 times duplication [i] based on digital (144, 171, 1517)-net over F3, using
- net defined by OOA [i] based on linear OOA(3171, 1517, F3, 27, 27) (dual of [(1517, 27), 40788, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3171, 19722, F3, 27) (dual of [19722, 19551, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3171, 19724, F3, 27) (dual of [19724, 19553, 28]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3171, 19724, F3, 27) (dual of [19724, 19553, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3171, 19722, F3, 27) (dual of [19722, 19551, 28]-code), using
- net defined by OOA [i] based on linear OOA(3171, 1517, F3, 27, 27) (dual of [(1517, 27), 40788, 28]-NRT-code), using
(147, 147+27, 10169)-Net over F3 — Digital
Digital (147, 174, 10169)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3174, 10169, F3, 27) (dual of [10169, 9995, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3174, 19725, F3, 27) (dual of [19725, 19551, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(3163, 19684, F3, 27) (dual of [19684, 19521, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3127, 19684, F3, 21) (dual of [19684, 19557, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(311, 41, F3, 5) (dual of [41, 30, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- (u, u+v)-construction [i] based on
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3174, 19725, F3, 27) (dual of [19725, 19551, 28]-code), using
(147, 147+27, 6335168)-Net in Base 3 — Upper bound on s
There is no (147, 174, 6335169)-net in base 3, because
- 1 times m-reduction [i] would yield (147, 173, 6335169)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 34831 921244 675151 407505 624818 536007 243432 359989 298189 824366 902837 108264 039870 698835 > 3173 [i]