Best Known (73, 73+13, s)-Nets in Base 27
(73, 73+13, 1575978)-Net over F27 — Constructive and digital
Digital (73, 86, 1575978)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (19, 25, 177878)-net over F27, using
- (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 (15, 21, 177148)-net over F27, using
- net defined by OOA [i] based on linear OOA(2721, 177148, F27, 6, 6) (dual of [(177148, 6), 1062867, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2721, 531444, F27, 6) (dual of [531444, 531423, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(2721, 531445, F27, 6) (dual of [531445, 531424, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(2721, 531441, F27, 6) (dual of [531441, 531420, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2717, 531441, F27, 5) (dual of [531441, 531424, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(270, 4, F27, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(2721, 531445, F27, 6) (dual of [531445, 531424, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(2721, 531444, F27, 6) (dual of [531444, 531423, 7]-code), using
- net defined by OOA [i] based on linear OOA(2721, 177148, F27, 6, 6) (dual of [(177148, 6), 1062867, 7]-NRT-code), using
- digital (1, 4, 730)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (48, 61, 1398100)-net over F27, using
- net defined by OOA [i] based on linear OOA(2761, 1398100, F27, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2761, 8388601, F27, 13) (dual of [8388601, 8388540, 14]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2761, 8388601, F27, 13) (dual of [8388601, 8388540, 14]-code), using
- net defined by OOA [i] based on linear OOA(2761, 1398100, F27, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- digital (19, 25, 177878)-net over F27, using
(73, 73+13, large)-Net over F27 — Digital
Digital (73, 86, large)-net over F27, using
- t-expansion [i] based on digital (72, 86, large)-net over F27, using
- 5 times m-reduction [i] based on digital (72, 91, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2791, large, F27, 19) (dual of [large, large−91, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2791, large, F27, 19) (dual of [large, large−91, 20]-code), using
- 5 times m-reduction [i] based on digital (72, 91, large)-net over F27, using
(73, 73+13, large)-Net in Base 27 — Upper bound on s
There is no (73, 86, large)-net in base 27, because
- 11 times m-reduction [i] would yield (73, 75, large)-net in base 27, but