Best Known (167, 167+13, s)-Nets in Base 3
(167, 167+13, 2796200)-Net over F3 — Constructive and digital
Digital (167, 180, 2796200)-net over F3, using
- 32 times duplication [i] based on digital (165, 178, 2796200)-net over F3, using
- net defined by OOA [i] based on linear OOA(3178, 2796200, F3, 14, 13) (dual of [(2796200, 14), 39146622, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3178, 8388601, F3, 2, 13) (dual of [(8388601, 2), 16777024, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3178, 8388602, F3, 2, 13) (dual of [(8388602, 2), 16777026, 14]-NRT-code), using
- trace code [i] based on linear OOA(989, 4194301, F9, 2, 13) (dual of [(4194301, 2), 8388513, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(989, 8388602, F9, 13) (dual of [8388602, 8388513, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(989, large, F9, 13) (dual of [large, large−89, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(989, large, F9, 13) (dual of [large, large−89, 14]-code), using
- OOA 2-folding [i] based on linear OA(989, 8388602, F9, 13) (dual of [8388602, 8388513, 14]-code), using
- trace code [i] based on linear OOA(989, 4194301, F9, 2, 13) (dual of [(4194301, 2), 8388513, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3178, 8388602, F3, 2, 13) (dual of [(8388602, 2), 16777026, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3178, 8388601, F3, 2, 13) (dual of [(8388601, 2), 16777024, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(3178, 2796200, F3, 14, 13) (dual of [(2796200, 14), 39146622, 14]-NRT-code), using
(167, 167+13, large)-Net over F3 — Digital
Digital (167, 180, large)-net over F3, using
- 32 times duplication [i] based on digital (165, 178, large)-net over F3, using
- t-expansion [i] based on digital (163, 178, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3178, large, F3, 15) (dual of [large, large−178, 16]-code), using
- 28 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]
- 28 times code embedding in larger space [i] based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3178, large, F3, 15) (dual of [large, large−178, 16]-code), using
- t-expansion [i] based on digital (163, 178, large)-net over F3, using
(167, 167+13, large)-Net in Base 3 — Upper bound on s
There is no (167, 180, large)-net in base 3, because
- 11 times m-reduction [i] would yield (167, 169, large)-net in base 3, but