Best Known (192, 192+25, s)-Nets in Base 3
(192, 192+25, 132863)-Net over F3 — Constructive and digital
Digital (192, 217, 132863)-net over F3, using
- 31 times duplication [i] based on digital (191, 216, 132863)-net over F3, using
- net defined by OOA [i] based on linear OOA(3216, 132863, F3, 25, 25) (dual of [(132863, 25), 3321359, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3216, 1594357, F3, 25) (dual of [1594357, 1594141, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3215, 1594356, F3, 25) (dual of [1594356, 1594141, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(3209, 1594324, F3, 25) (dual of [1594324, 1594115, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(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([0,12]) ⊂ C([0,10]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3215, 1594356, F3, 25) (dual of [1594356, 1594141, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3216, 1594357, F3, 25) (dual of [1594357, 1594141, 26]-code), using
- net defined by OOA [i] based on linear OOA(3216, 132863, F3, 25, 25) (dual of [(132863, 25), 3321359, 26]-NRT-code), using
(192, 192+25, 398591)-Net over F3 — Digital
Digital (192, 217, 398591)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3217, 398591, F3, 4, 25) (dual of [(398591, 4), 1594147, 26]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3217, 1594364, F3, 25) (dual of [1594364, 1594147, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3170, 1594323, F3, 20) (dual of [1594323, 1594153, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(24) ⊂ Ce(19) [i] based on
- OOA 4-folding [i] based on linear OA(3217, 1594364, F3, 25) (dual of [1594364, 1594147, 26]-code), using
(192, 192+25, large)-Net in Base 3 — Upper bound on s
There is no (192, 217, large)-net in base 3, because
- 23 times m-reduction [i] would yield (192, 194, large)-net in base 3, but