Best Known (161, 171, s)-Nets in Base 3
(161, 171, 6711977)-Net over F3 — Constructive and digital
Digital (161, 171, 6711977)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (18, 23, 1097)-net over F3, using
- net defined by OOA [i] based on linear OOA(323, 1097, F3, 5, 5) (dual of [(1097, 5), 5462, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(323, 2195, F3, 5) (dual of [2195, 2172, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(323, 2196, F3, 5) (dual of [2196, 2173, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(322, 2187, F3, 5) (dual of [2187, 2165, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(38, 9, F3, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,3)), using
- dual of repetition code with length 9 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 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 X4 applied to Ce(4) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(323, 2196, F3, 5) (dual of [2196, 2173, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(323, 2195, F3, 5) (dual of [2195, 2172, 6]-code), using
- net defined by OOA [i] based on linear OOA(323, 1097, F3, 5, 5) (dual of [(1097, 5), 5462, 6]-NRT-code), using
- digital (138, 148, 6710880)-net over F3, using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F81, using
- net defined by OOA [i] based on linear OOA(8137, 1677720, F81, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(8137, 8388600, F81, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(8137, large, F81, 10) (dual of [large, large−37, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(8137, large, F81, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(8137, 8388600, F81, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(8137, 1677720, F81, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F81, using
- digital (18, 23, 1097)-net over F3, using
(161, 171, large)-Net over F3 — Digital
Digital (161, 171, large)-net over F3, using
- 36 times duplication [i] based on digital (155, 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
(161, 171, large)-Net in Base 3 — Upper bound on s
There is no (161, 171, large)-net in base 3, because
- 8 times m-reduction [i] would yield (161, 163, large)-net in base 3, but