Best Known (17, 26, s)-Nets in Base 27
(17, 26, 4922)-Net over F27 — Constructive and digital
Digital (17, 26, 4922)-net over F27, using
- net defined by OOA [i] based on linear OOA(2726, 4922, F27, 9, 9) (dual of [(4922, 9), 44272, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2726, 19689, F27, 9) (dual of [19689, 19663, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2726, 19691, F27, 9) (dual of [19691, 19665, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(2725, 19684, F27, 9) (dual of [19684, 19659, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(2719, 19684, F27, 7) (dual of [19684, 19665, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 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([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2726, 19691, F27, 9) (dual of [19691, 19665, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2726, 19689, F27, 9) (dual of [19689, 19663, 10]-code), using
(17, 26, 16822)-Net over F27 — Digital
Digital (17, 26, 16822)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2726, 16822, F27, 9) (dual of [16822, 16796, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2726, 19691, F27, 9) (dual of [19691, 19665, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(2725, 19684, F27, 9) (dual of [19684, 19659, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(2719, 19684, F27, 7) (dual of [19684, 19665, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 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([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2726, 19691, F27, 9) (dual of [19691, 19665, 10]-code), using
(17, 26, large)-Net in Base 27 — Upper bound on s
There is no (17, 26, large)-net in base 27, because
- 7 times m-reduction [i] would yield (17, 19, large)-net in base 27, but