Best Known (61, 61+10, s)-Nets in Base 27
(61, 61+10, 3356170)-Net over F27 — Constructive and digital
Digital (61, 71, 3356170)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (1, 4, 730)-net over F27, using
- net defined by OOA [i] based on linear OOA(274, 730, F27, 3, 3) (dual of [(730, 3), 2186, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(274, 730, F27, 2, 3) (dual of [(730, 2), 1456, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(274, 730, F27, 3, 3) (dual of [(730, 3), 2186, 4]-NRT-code), using
- digital (16, 21, 1677720)-net over F27, using
- s-reduction based on digital (16, 21, 4194301)-net over F27, using
- net defined by OOA [i] based on linear OOA(2721, 4194301, F27, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2721, large, F27, 5) (dual of [large, large−21, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(2721, large, F27, 5) (dual of [large, large−21, 6]-code), using
- net defined by OOA [i] based on linear OOA(2721, 4194301, F27, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
- s-reduction based on digital (16, 21, 4194301)-net over F27, 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 (1, 4, 730)-net over F27, using
(61, 61+10, large)-Net over F27 — Digital
Digital (61, 71, large)-net over F27, using
- t-expansion [i] based on digital (60, 71, large)-net over F27, using
- 5 times m-reduction [i] based on digital (60, 76, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2776, large, F27, 16) (dual of [large, large−76, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2776, large, F27, 16) (dual of [large, large−76, 17]-code), using
- 5 times m-reduction [i] based on digital (60, 76, large)-net over F27, using
(61, 61+10, large)-Net in Base 27 — Upper bound on s
There is no (61, 71, large)-net in base 27, because
- 8 times m-reduction [i] would yield (61, 63, large)-net in base 27, but