Best Known (162, 162+13, s)-Nets in Base 3
(162, 162+13, 1929550)-Net over F3 — Constructive and digital
Digital (162, 175, 1929550)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (48, 54, 531450)-net over F3, using
- net defined by OOA [i] based on linear OOA(354, 531450, F3, 6, 6) (dual of [(531450, 6), 3188646, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(354, 531450, F3, 5, 6) (dual of [(531450, 5), 2657196, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(354, 1594350, F3, 6) (dual of [1594350, 1594296, 7]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(353, 1594323, F3, 7) (dual of [1594323, 1594270, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(327, 1594323, F3, 4) (dual of [1594323, 1594296, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(31, 27, F3, 1) (dual of [27, 26, 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(354, 1594350, F3, 6) (dual of [1594350, 1594296, 7]-code), using
- appending kth column [i] based on linear OOA(354, 531450, F3, 5, 6) (dual of [(531450, 5), 2657196, 7]-NRT-code), using
- net defined by OOA [i] based on linear OOA(354, 531450, F3, 6, 6) (dual of [(531450, 6), 3188646, 7]-NRT-code), using
- digital (108, 121, 1398100)-net over F3, using
- net defined by OOA [i] based on linear OOA(3121, 1398100, F3, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3121, 8388601, F3, 13) (dual of [8388601, 8388480, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, 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(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3121, 8388601, F3, 13) (dual of [8388601, 8388480, 14]-code), using
- net defined by OOA [i] based on linear OOA(3121, 1398100, F3, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
- digital (48, 54, 531450)-net over F3, using
(162, 162+13, large)-Net over F3 — Digital
Digital (162, 175, large)-net over F3, using
- 310 times duplication [i] based on digital (152, 165, large)-net over F3, using
- t-expansion [i] based on digital (151, 165, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3165, large, F3, 14) (dual of [large, large−165, 15]-code), using
- 29 times code embedding in larger space [i] based on linear OA(3136, large, F3, 14) (dual of [large, large−136, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 29 times code embedding in larger space [i] based on linear OA(3136, large, F3, 14) (dual of [large, large−136, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3165, large, F3, 14) (dual of [large, large−165, 15]-code), using
- t-expansion [i] based on digital (151, 165, large)-net over F3, using
(162, 162+13, large)-Net in Base 3 — Upper bound on s
There is no (162, 175, large)-net in base 3, because
- 11 times m-reduction [i] would yield (162, 164, large)-net in base 3, but