Best Known (7, 14, s)-Nets in Base 27
(7, 14, 244)-Net over F27 — Constructive and digital
Digital (7, 14, 244)-net over F27, using
- net defined by OOA [i] based on linear OOA(2714, 244, F27, 7, 7) (dual of [(244, 7), 1694, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2714, 733, F27, 7) (dual of [733, 719, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2714, 735, F27, 7) (dual of [735, 721, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(2713, 730, F27, 7) (dual of [730, 717, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(279, 730, F27, 5) (dual of [730, 721, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 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([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2714, 735, F27, 7) (dual of [735, 721, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2714, 733, F27, 7) (dual of [733, 719, 8]-code), using
(7, 14, 526)-Net over F27 — Digital
Digital (7, 14, 526)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2714, 526, F27, 7) (dual of [526, 512, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2714, 735, F27, 7) (dual of [735, 721, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(2713, 730, F27, 7) (dual of [730, 717, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(279, 730, F27, 5) (dual of [730, 721, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 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([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2714, 735, F27, 7) (dual of [735, 721, 8]-code), using
(7, 14, 111424)-Net in Base 27 — Upper bound on s
There is no (7, 14, 111425)-net in base 27, because
- 1 times m-reduction [i] would yield (7, 13, 111425)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 4 052562 895485 268451 > 2713 [i]