Best Known (58−10, 58, s)-Nets in Base 27
(58−10, 58, 1679125)-Net over F27 — Constructive and digital
Digital (48, 58, 1679125)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 12, 1405)-net over F27, using
- net defined by OOA [i] based on linear OOA(2712, 1405, F27, 5, 5) (dual of [(1405, 5), 7013, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2712, 2811, F27, 5) (dual of [2811, 2799, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(2712, 2812, F27, 5) (dual of [2812, 2800, 6]-code), using
- generalized (u, u+v)-construction [i] based on
- linear OA(271, 703, F27, 1) (dual of [703, 702, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(271, 703, F27, 1) (dual of [703, 702, 2]-code) (see above)
- linear OA(273, 703, F27, 2) (dual of [703, 700, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(273, 728, F27, 2) (dual of [728, 725, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(273, 728, F27, 2) (dual of [728, 725, 3]-code), using
- linear OA(277, 703, F27, 5) (dual of [703, 696, 6]-code), using
- linear OA(271, 703, F27, 1) (dual of [703, 702, 2]-code), using
- generalized (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(2712, 2812, F27, 5) (dual of [2812, 2800, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2712, 2811, F27, 5) (dual of [2811, 2799, 6]-code), using
- net defined by OOA [i] based on linear OOA(2712, 1405, F27, 5, 5) (dual of [(1405, 5), 7013, 6]-NRT-code), using
- digital (36, 46, 1677720)-net over F27, using
- net defined by OOA [i] based on linear OOA(2746, 1677720, F27, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2746, 8388600, F27, 10) (dual of [8388600, 8388554, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(2746, large, F27, 10) (dual of [large, large−46, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(2746, large, F27, 10) (dual of [large, large−46, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(2746, 8388600, F27, 10) (dual of [8388600, 8388554, 11]-code), using
- net defined by OOA [i] based on linear OOA(2746, 1677720, F27, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- digital (7, 12, 1405)-net over F27, using
(58−10, 58, 1681042)-Net in Base 27 — Constructive
(48, 58, 1681042)-net in base 27, using
- (u, u+v)-construction [i] based on
- (7, 12, 3322)-net in base 27, using
- base change [i] based on digital (4, 9, 3322)-net over F81, using
- net defined by OOA [i] based on linear OOA(819, 3322, F81, 5, 5) (dual of [(3322, 5), 16601, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(819, 3322, F81, 4, 5) (dual of [(3322, 4), 13279, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(812, 82, F81, 4, 2) (dual of [(82, 4), 326, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;326,81) [i]
- linear OOA(817, 3240, F81, 4, 5) (dual of [(3240, 4), 12953, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(817, 6481, F81, 5) (dual of [6481, 6474, 6]-code), using
- linear OOA(812, 82, F81, 4, 2) (dual of [(82, 4), 326, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(819, 3322, F81, 4, 5) (dual of [(3322, 4), 13279, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(819, 3322, F81, 5, 5) (dual of [(3322, 5), 16601, 6]-NRT-code), using
- base change [i] based on digital (4, 9, 3322)-net over F81, using
- digital (36, 46, 1677720)-net over F27, using
- net defined by OOA [i] based on linear OOA(2746, 1677720, F27, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2746, 8388600, F27, 10) (dual of [8388600, 8388554, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(2746, large, F27, 10) (dual of [large, large−46, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(2746, large, F27, 10) (dual of [large, large−46, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(2746, 8388600, F27, 10) (dual of [8388600, 8388554, 11]-code), using
- net defined by OOA [i] based on linear OOA(2746, 1677720, F27, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- (7, 12, 3322)-net in base 27, using
(58−10, 58, large)-Net over F27 — Digital
Digital (48, 58, large)-net over F27, using
- 3 times m-reduction [i] based on digital (48, 61, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
(58−10, 58, large)-Net in Base 27 — Upper bound on s
There is no (48, 58, large)-net in base 27, because
- 8 times m-reduction [i] would yield (48, 50, large)-net in base 27, but