Best Known (83, 83+14, s)-Nets in Base 27
(83, 83+14, 2396742)-Net over F27 — Constructive and digital
Digital (83, 97, 2396742)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (24, 31, 2796200)-net over F27, using
- net defined by OOA [i] based on linear OOA(2731, 2796200, F27, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2731, 8388601, F27, 7) (dual of [8388601, 8388570, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2731, large, F27, 7) (dual of [large, large−31, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2731, large, F27, 7) (dual of [large, large−31, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2731, 8388601, F27, 7) (dual of [8388601, 8388570, 8]-code), using
- net defined by OOA [i] based on linear OOA(2731, 2796200, F27, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- digital (52, 66, 1198371)-net over F27, using
- net defined by OOA [i] based on linear OOA(2766, 1198371, F27, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2766, 8388597, F27, 14) (dual of [8388597, 8388531, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2766, large, F27, 14) (dual of [large, large−66, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(2766, large, F27, 14) (dual of [large, large−66, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(2766, 8388597, F27, 14) (dual of [8388597, 8388531, 15]-code), using
- net defined by OOA [i] based on linear OOA(2766, 1198371, F27, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
- digital (24, 31, 2796200)-net over F27, using
(83, 83+14, large)-Net over F27 — Digital
Digital (83, 97, large)-net over F27, using
- t-expansion [i] based on digital (80, 97, large)-net over F27, using
- 4 times m-reduction [i] based on digital (80, 101, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27101, large, F27, 21) (dual of [large, large−101, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27101, large, F27, 21) (dual of [large, large−101, 22]-code), using
- 4 times m-reduction [i] based on digital (80, 101, large)-net over F27, using
(83, 83+14, large)-Net in Base 27 — Upper bound on s
There is no (83, 97, large)-net in base 27, because
- 12 times m-reduction [i] would yield (83, 85, large)-net in base 27, but