Best Known (228−31, 228, s)-Nets in Base 3
(228−31, 228, 11811)-Net over F3 — Constructive and digital
Digital (197, 228, 11811)-net over F3, using
- 33 times duplication [i] based on digital (194, 225, 11811)-net over F3, using
- net defined by OOA [i] based on linear OOA(3225, 11811, F3, 31, 31) (dual of [(11811, 31), 365916, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3225, 177166, F3, 31) (dual of [177166, 176941, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3225, 177173, F3, 31) (dual of [177173, 176948, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(3221, 177147, F3, 31) (dual of [177147, 176926, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(34, 26, F3, 2) (dual of [26, 22, 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(30) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3225, 177173, F3, 31) (dual of [177173, 176948, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3225, 177166, F3, 31) (dual of [177166, 176941, 32]-code), using
- net defined by OOA [i] based on linear OOA(3225, 11811, F3, 31, 31) (dual of [(11811, 31), 365916, 32]-NRT-code), using
(228−31, 228, 51662)-Net over F3 — Digital
Digital (197, 228, 51662)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3228, 51662, F3, 3, 31) (dual of [(51662, 3), 154758, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3228, 59059, F3, 3, 31) (dual of [(59059, 3), 176949, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3228, 177177, F3, 31) (dual of [177177, 176949, 32]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3227, 177176, F3, 31) (dual of [177176, 176949, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(3221, 177148, F3, 31) (dual of [177148, 176927, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3199, 177148, F3, 27) (dual of [177148, 176949, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-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([0,15]) ⊂ C([0,13]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3227, 177176, F3, 31) (dual of [177176, 176949, 32]-code), using
- OOA 3-folding [i] based on linear OA(3228, 177177, F3, 31) (dual of [177177, 176949, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(3228, 59059, F3, 3, 31) (dual of [(59059, 3), 176949, 32]-NRT-code), using
(228−31, 228, large)-Net in Base 3 — Upper bound on s
There is no (197, 228, large)-net in base 3, because
- 29 times m-reduction [i] would yield (197, 199, large)-net in base 3, but