Best Known (170, 170+11, s)-Nets in Base 3
(170, 170+11, 6711111)-Net over F3 — Constructive and digital
Digital (170, 181, 6711111)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (12, 17, 231)-net over F3, using
- net defined by OOA [i] based on linear OOA(317, 231, F3, 5, 5) (dual of [(231, 5), 1138, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(317, 231, F3, 4, 5) (dual of [(231, 4), 907, 6]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(31, 77, F3, 4, 1) (dual of [(77, 4), 307, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(31, s, F3, 4, 1) with arbitrarily large s, using
- appending 3 arbitrary columns [i] based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- discarding factors / shortening the dual code based on linear OOA(31, s, F3, 4, 1) with arbitrarily large s, using
- linear OOA(35, 77, F3, 4, 2) (dual of [(77, 4), 303, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(35, 121, F3, 4, 2) (dual of [(121, 4), 479, 3]-NRT-code), using
- appending 2 arbitrary columns [i] based on linear OOA(35, 121, F3, 2, 2) (dual of [(121, 2), 237, 3]-NRT-code), using
- appending kth column [i] based on linear OA(35, 121, F3, 2) (dual of [121, 116, 3]-code), using
- Hamming code H(5,3) [i]
- appending kth column [i] based on linear OA(35, 121, F3, 2) (dual of [121, 116, 3]-code), using
- appending 2 arbitrary columns [i] based on linear OOA(35, 121, F3, 2, 2) (dual of [(121, 2), 237, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(35, 121, F3, 4, 2) (dual of [(121, 4), 479, 3]-NRT-code), using
- linear OOA(311, 77, F3, 4, 5) (dual of [(77, 4), 297, 6]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 11, 77)-net over F3, using
- linear OOA(31, 77, F3, 4, 1) (dual of [(77, 4), 307, 2]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(317, 231, F3, 4, 5) (dual of [(231, 4), 907, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(317, 231, F3, 5, 5) (dual of [(231, 5), 1138, 6]-NRT-code), using
- digital (153, 164, 6710880)-net over F3, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F81, using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F81, using
- digital (12, 17, 231)-net over F3, using
(170, 170+11, large)-Net over F3 — Digital
Digital (170, 181, large)-net over F3, using
- 33 times duplication [i] based on digital (167, 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
(170, 170+11, large)-Net in Base 3 — Upper bound on s
There is no (170, 181, large)-net in base 3, because
- 9 times m-reduction [i] would yield (170, 172, large)-net in base 3, but