Best Known (199, 199+12, s)-Nets in Base 3
(199, 199+12, 5593134)-Net over F3 — Constructive and digital
Digital (199, 211, 5593134)-net over F3, using
- 31 times duplication [i] based on digital (198, 210, 5593134)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (24, 30, 734)-net over F3, using
- net defined by OOA [i] based on linear OOA(330, 734, F3, 6, 6) (dual of [(734, 6), 4374, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(330, 734, F3, 5, 6) (dual of [(734, 5), 3640, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(330, 2202, F3, 6) (dual of [2202, 2172, 7]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(329, 2187, F3, 7) (dual of [2187, 2158, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(315, 2187, F3, 4) (dual of [2187, 2172, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(330, 2202, F3, 6) (dual of [2202, 2172, 7]-code), using
- appending kth column [i] based on linear OOA(330, 734, F3, 5, 6) (dual of [(734, 5), 3640, 7]-NRT-code), using
- net defined by OOA [i] based on linear OOA(330, 734, F3, 6, 6) (dual of [(734, 6), 4374, 7]-NRT-code), using
- digital (168, 180, 5592400)-net over F3, using
- trace code for nets [i] based on digital (78, 90, 2796200)-net over F9, using
- net defined by OOA [i] based on linear OOA(990, 2796200, F9, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(990, 8388601, F9, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(990, 8388602, F9, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- trace code [i] based on linear OOA(8145, 4194301, F81, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8145, 8388602, F81, 12) (dual of [8388602, 8388557, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- OOA 2-folding [i] based on linear OA(8145, 8388602, F81, 12) (dual of [8388602, 8388557, 13]-code), using
- trace code [i] based on linear OOA(8145, 4194301, F81, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(990, 8388602, F9, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(990, 8388601, F9, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(990, 2796200, F9, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- trace code for nets [i] based on digital (78, 90, 2796200)-net over F9, using
- digital (24, 30, 734)-net over F3, using
- (u, u+v)-construction [i] based on
(199, 199+12, large)-Net over F3 — Digital
Digital (199, 211, large)-net over F3, using
- t-expansion [i] based on digital (198, 211, large)-net over F3, using
- 5 times m-reduction [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
- 5 times m-reduction [i] based on digital (198, 216, large)-net over F3, using
(199, 199+12, large)-Net in Base 3 — Upper bound on s
There is no (199, 211, large)-net in base 3, because
- 10 times m-reduction [i] would yield (199, 201, large)-net in base 3, but