Best Known (47, 47+15, s)-Nets in Base 27
(47, 47+15, 75924)-Net over F27 — Constructive and digital
Digital (47, 62, 75924)-net over F27, using
- net defined by OOA [i] based on linear OOA(2762, 75924, F27, 15, 15) (dual of [(75924, 15), 1138798, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2762, 531469, F27, 15) (dual of [531469, 531407, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2762, 531470, F27, 15) (dual of [531470, 531408, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- linear OA(2757, 531442, F27, 15) (dual of [531442, 531385, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(2733, 531442, F27, 9) (dual of [531442, 531409, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(275, 28, F27, 5) (dual of [28, 23, 6]-code or 28-arc in PG(4,27)), using
- extended Reed–Solomon code RSe(23,27) [i]
- the expurgated narrow-sense BCH-code C(I) with length 28 | 272−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2762, 531470, F27, 15) (dual of [531470, 531408, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2762, 531469, F27, 15) (dual of [531469, 531407, 16]-code), using
(47, 47+15, 531470)-Net over F27 — Digital
Digital (47, 62, 531470)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2762, 531470, F27, 15) (dual of [531470, 531408, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- linear OA(2757, 531442, F27, 15) (dual of [531442, 531385, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(2733, 531442, F27, 9) (dual of [531442, 531409, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(275, 28, F27, 5) (dual of [28, 23, 6]-code or 28-arc in PG(4,27)), using
- extended Reed–Solomon code RSe(23,27) [i]
- the expurgated narrow-sense BCH-code C(I) with length 28 | 272−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
(47, 47+15, large)-Net in Base 27 — Upper bound on s
There is no (47, 62, large)-net in base 27, because
- 13 times m-reduction [i] would yield (47, 49, large)-net in base 27, but