Best Known (69, 69+11, s)-Nets in Base 27
(69, 69+11, 4600573)-Net over F27 — Constructive and digital
Digital (69, 80, 4600573)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (5, 8, 1245133)-net over F27, using
- net defined by OOA [i] based on linear OOA(278, 1245133, F27, 3, 3) (dual of [(1245133, 3), 3735391, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(278, 1245133, F27, 2, 3) (dual of [(1245133, 2), 2490258, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(278, 1245133, F27, 3) (dual of [1245133, 1245125, 4]-code or 1245133-cap in PG(7,27)), using
- appending kth column [i] based on linear OOA(278, 1245133, F27, 2, 3) (dual of [(1245133, 2), 2490258, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(278, 1245133, F27, 3, 3) (dual of [(1245133, 3), 3735391, 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 (40, 51, 1677720)-net over F27, using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, 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(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- digital (5, 8, 1245133)-net over F27, using
(69, 69+11, large)-Net over F27 — Digital
Digital (69, 80, large)-net over F27, using
- t-expansion [i] based on digital (68, 80, large)-net over F27, using
- 6 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
- 6 times m-reduction [i] based on digital (68, 86, large)-net over F27, using
(69, 69+11, large)-Net in Base 27 — Upper bound on s
There is no (69, 80, large)-net in base 27, because
- 9 times m-reduction [i] would yield (69, 71, large)-net in base 27, but