Best Known (51−17, 51, s)-Nets in Base 27
(51−17, 51, 2461)-Net over F27 — Constructive and digital
Digital (34, 51, 2461)-net over F27, using
- 271 times duplication [i] based on digital (33, 50, 2461)-net over F27, using
- net defined by OOA [i] based on linear OOA(2750, 2461, F27, 17, 17) (dual of [(2461, 17), 41787, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2750, 19689, F27, 17) (dual of [19689, 19639, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2750, 19691, F27, 17) (dual of [19691, 19641, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(2749, 19684, F27, 17) (dual of [19684, 19635, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2743, 19684, F27, 15) (dual of [19684, 19641, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,8]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2750, 19691, F27, 17) (dual of [19691, 19641, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2750, 19689, F27, 17) (dual of [19689, 19639, 18]-code), using
- net defined by OOA [i] based on linear OOA(2750, 2461, F27, 17, 17) (dual of [(2461, 17), 41787, 18]-NRT-code), using
(51−17, 51, 14581)-Net over F27 — Digital
Digital (34, 51, 14581)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2751, 14581, F27, 17) (dual of [14581, 14530, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2751, 19694, F27, 17) (dual of [19694, 19643, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(2749, 19683, F27, 17) (dual of [19683, 19634, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2740, 19683, F27, 14) (dual of [19683, 19643, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(2751, 19694, F27, 17) (dual of [19694, 19643, 18]-code), using
(51−17, 51, large)-Net in Base 27 — Upper bound on s
There is no (34, 51, large)-net in base 27, because
- 15 times m-reduction [i] would yield (34, 36, large)-net in base 27, but