Best Known (226−13, 226, s)-Nets in Base 3
(226−13, 226, 5593134)-Net over F3 — Constructive and digital
Digital (213, 226, 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 (183, 196, 5592400)-net over F3, using
- trace code for nets [i] based on digital (85, 98, 2796200)-net over F9, using
- net defined by OOA [i] based on linear OOA(998, 2796200, F9, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(998, 8388601, F9, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(998, 8388602, F9, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- trace code [i] based on linear OOA(8149, 4194301, F81, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8149, 8388602, F81, 13) (dual of [8388602, 8388553, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- OOA 2-folding [i] based on linear OA(8149, 8388602, F81, 13) (dual of [8388602, 8388553, 14]-code), using
- trace code [i] based on linear OOA(8149, 4194301, F81, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(998, 8388602, F9, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(998, 8388601, F9, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(998, 2796200, F9, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- trace code for nets [i] based on digital (85, 98, 2796200)-net over F9, using
- digital (24, 30, 734)-net over F3, using
(226−13, 226, large)-Net over F3 — Digital
Digital (213, 226, large)-net over F3, using
- t-expansion [i] based on digital (209, 226, large)-net over F3, using
- 2 times m-reduction [i] based on digital (209, 228, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3228, large, F3, 19) (dual of [large, large−228, 20]-code), using
- 47 times code embedding in larger space [i] based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 47 times code embedding in larger space [i] based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3228, large, F3, 19) (dual of [large, large−228, 20]-code), using
- 2 times m-reduction [i] based on digital (209, 228, large)-net over F3, using
(226−13, 226, large)-Net in Base 3 — Upper bound on s
There is no (213, 226, large)-net in base 3, because
- 11 times m-reduction [i] would yield (213, 215, large)-net in base 3, but