Best Known (89−21, 89, s)-Nets in Base 27
(89−21, 89, 53147)-Net over F27 — Constructive and digital
Digital (68, 89, 53147)-net over F27, using
- 272 times duplication [i] based on digital (66, 87, 53147)-net over F27, using
- net defined by OOA [i] based on linear OOA(2787, 53147, F27, 21, 21) (dual of [(53147, 21), 1116000, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2787, 531471, F27, 21) (dual of [531471, 531384, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2787, 531472, F27, 21) (dual of [531472, 531385, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(2781, 531442, F27, 21) (dual of [531442, 531361, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 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(276, 30, F27, 5) (dual of [30, 24, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([1,2]) [i] based on
- linear OA(275, 28, F27, 5) (dual of [28, 23, 6]-code or 28-arc in PG(4,27)), using 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]
- linear OA(274, 28, F27, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,27)), using the narrow-sense BCH-code C(I) with length 28 | 272−1, defining interval I = [1,2], and minimum distance d ≥ |{1,2}| + |{−3,0}| = 4 (simple Roos-bound) [i]
- linear OA(271, 2, F27, 1) (dual of [2, 1, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,2]) ⊂ C([1,2]) [i] based on
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2787, 531472, F27, 21) (dual of [531472, 531385, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2787, 531471, F27, 21) (dual of [531471, 531384, 22]-code), using
- net defined by OOA [i] based on linear OOA(2787, 53147, F27, 21, 21) (dual of [(53147, 21), 1116000, 22]-NRT-code), using
(89−21, 89, 748010)-Net over F27 — Digital
Digital (68, 89, 748010)-net over F27, using
(89−21, 89, large)-Net in Base 27 — Upper bound on s
There is no (68, 89, large)-net in base 27, because
- 19 times m-reduction [i] would yield (68, 70, large)-net in base 27, but