Best Known (177−15, 177, s)-Nets in Base 3
(177−15, 177, 1198618)-Net over F3 — Constructive and digital
Digital (162, 177, 1198618)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (20, 27, 247)-net over F3, using
- net defined by OOA [i] based on linear OOA(327, 247, F3, 7, 7) (dual of [(247, 7), 1702, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(327, 742, F3, 7) (dual of [742, 715, 8]-code), using
- construction XX applied to C1 = C([363,367]), C2 = C([361,365]), C3 = C1 + C2 = C([363,365]), and C∩ = C1 ∩ C2 = C([361,367]) [i] based on
- linear OA(319, 728, F3, 5) (dual of [728, 709, 6]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {363,364,365,366,367}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(319, 728, F3, 5) (dual of [728, 709, 6]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {361,362,363,364,365}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(325, 728, F3, 7) (dual of [728, 703, 8]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {361,362,…,367}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(313, 728, F3, 3) (dual of [728, 715, 4]-code or 728-cap in PG(12,3)), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {363,364,365}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(31, 7, F3, 1) (dual of [7, 6, 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
- linear OA(31, 7, F3, 1) (dual of [7, 6, 2]-code) (see above)
- construction XX applied to C1 = C([363,367]), C2 = C([361,365]), C3 = C1 + C2 = C([363,365]), and C∩ = C1 ∩ C2 = C([361,367]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(327, 742, F3, 7) (dual of [742, 715, 8]-code), using
- net defined by OOA [i] based on linear OOA(327, 247, F3, 7, 7) (dual of [(247, 7), 1702, 8]-NRT-code), using
- digital (135, 150, 1198371)-net over F3, using
- net defined by OOA [i] based on linear OOA(3150, 1198371, F3, 15, 15) (dual of [(1198371, 15), 17975415, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3150, 8388598, F3, 15) (dual of [8388598, 8388448, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3150, 8388598, F3, 15) (dual of [8388598, 8388448, 16]-code), using
- net defined by OOA [i] based on linear OOA(3150, 1198371, F3, 15, 15) (dual of [(1198371, 15), 17975415, 16]-NRT-code), using
- digital (20, 27, 247)-net over F3, using
(177−15, 177, 8163254)-Net over F3 — Digital
Digital (162, 177, 8163254)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3177, 8163254, F3, 15) (dual of [8163254, 8163077, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3177, large, F3, 15) (dual of [large, large−177, 16]-code), using
- 27 times code embedding in larger space [i] based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 27 times code embedding in larger space [i] based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3177, large, F3, 15) (dual of [large, large−177, 16]-code), using
(177−15, 177, large)-Net in Base 3 — Upper bound on s
There is no (162, 177, large)-net in base 3, because
- 13 times m-reduction [i] would yield (162, 164, large)-net in base 3, but