Best Known (222−18, 222, s)-Nets in Base 3
(222−18, 222, 932613)-Net over F3 — Constructive and digital
Digital (204, 222, 932613)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (33, 42, 546)-net over F3, using
- net defined by OOA [i] based on linear OOA(342, 546, F3, 9, 9) (dual of [(546, 9), 4872, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(342, 2185, F3, 9) (dual of [2185, 2143, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(342, 2186, F3, 9) (dual of [2186, 2144, 10]-code), using
- 1 times truncation [i] based on linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 1 times truncation [i] based on linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(342, 2186, F3, 9) (dual of [2186, 2144, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(342, 2185, F3, 9) (dual of [2185, 2143, 10]-code), using
- net defined by OOA [i] based on linear OOA(342, 546, F3, 9, 9) (dual of [(546, 9), 4872, 10]-NRT-code), using
- digital (162, 180, 932067)-net over F3, using
- net defined by OOA [i] based on linear OOA(3180, 932067, F3, 18, 18) (dual of [(932067, 18), 16777026, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- net defined by OOA [i] based on linear OOA(3180, 932067, F3, 18, 18) (dual of [(932067, 18), 16777026, 19]-NRT-code), using
- digital (33, 42, 546)-net over F3, using
(222−18, 222, large)-Net over F3 — Digital
Digital (204, 222, large)-net over F3, using
- 36 times duplication [i] based on digital (198, 216, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3216, large, F3, 18) (dual of [large, large−216, 19]-code), using
- 36 times code embedding in larger space [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 36 times code embedding in larger space [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3216, large, F3, 18) (dual of [large, large−216, 19]-code), using
(222−18, 222, large)-Net in Base 3 — Upper bound on s
There is no (204, 222, large)-net in base 3, because
- 16 times m-reduction [i] would yield (204, 206, large)-net in base 3, but