Best Known (83−12, 83, s)-Nets in Base 27
(83−12, 83, 2796200)-Net over F27 — Constructive and digital
Digital (71, 83, 2796200)-net over F27, using
- 271 times duplication [i] based on digital (70, 82, 2796200)-net over F27, using
- net defined by OOA [i] based on linear OOA(2782, 2796200, F27, 14, 12) (dual of [(2796200, 14), 39146718, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(2782, 8388601, F27, 2, 12) (dual of [(8388601, 2), 16777120, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2782, 8388602, F27, 2, 12) (dual of [(8388602, 2), 16777122, 13]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(2726, 4194301, F27, 2, 6) (dual of [(4194301, 2), 8388576, 7]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2726, 8388602, F27, 6) (dual of [8388602, 8388576, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(2726, large, F27, 6) (dual of [large, large−26, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(2726, large, F27, 6) (dual of [large, large−26, 7]-code), using
- OOA 2-folding [i] based on linear OA(2726, 8388602, F27, 6) (dual of [8388602, 8388576, 7]-code), using
- linear OOA(2756, 4194301, F27, 2, 12) (dual of [(4194301, 2), 8388546, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2756, 8388602, F27, 12) (dual of [8388602, 8388546, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2756, large, F27, 12) (dual of [large, large−56, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(2756, large, F27, 12) (dual of [large, large−56, 13]-code), using
- OOA 2-folding [i] based on linear OA(2756, 8388602, F27, 12) (dual of [8388602, 8388546, 13]-code), using
- linear OOA(2726, 4194301, F27, 2, 6) (dual of [(4194301, 2), 8388576, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2782, 8388602, F27, 2, 12) (dual of [(8388602, 2), 16777122, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(2782, 8388601, F27, 2, 12) (dual of [(8388601, 2), 16777120, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(2782, 2796200, F27, 14, 12) (dual of [(2796200, 14), 39146718, 13]-NRT-code), using
(83−12, 83, large)-Net over F27 — Digital
Digital (71, 83, large)-net over F27, using
- t-expansion [i] based on digital (68, 83, large)-net over F27, using
- 3 times m-reduction [i] based on digital (68, 86, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2786, large, F27, 18) (dual of [large, large−86, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2786, large, F27, 18) (dual of [large, large−86, 19]-code), using
- 3 times m-reduction [i] based on digital (68, 86, large)-net over F27, using
(83−12, 83, large)-Net in Base 27 — Upper bound on s
There is no (71, 83, large)-net in base 27, because
- 10 times m-reduction [i] would yield (71, 73, large)-net in base 27, but