Best Known (69−23, 69, s)-Nets in Base 27
(69−23, 69, 1790)-Net over F27 — Constructive and digital
Digital (46, 69, 1790)-net over F27, using
- 271 times duplication [i] based on digital (45, 68, 1790)-net over F27, using
- net defined by OOA [i] based on linear OOA(2768, 1790, F27, 23, 23) (dual of [(1790, 23), 41102, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2768, 19691, F27, 23) (dual of [19691, 19623, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(2767, 19684, F27, 23) (dual of [19684, 19617, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(2761, 19684, F27, 21) (dual of [19684, 19623, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,11]) ⊂ C([0,10]) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(2768, 19691, F27, 23) (dual of [19691, 19623, 24]-code), using
- net defined by OOA [i] based on linear OOA(2768, 1790, F27, 23, 23) (dual of [(1790, 23), 41102, 24]-NRT-code), using
(69−23, 69, 14391)-Net over F27 — Digital
Digital (46, 69, 14391)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2769, 14391, F27, 23) (dual of [14391, 14322, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2769, 19694, F27, 23) (dual of [19694, 19625, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(2767, 19683, F27, 23) (dual of [19683, 19616, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2758, 19683, F27, 20) (dual of [19683, 19625, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(2769, 19694, F27, 23) (dual of [19694, 19625, 24]-code), using
(69−23, 69, large)-Net in Base 27 — Upper bound on s
There is no (46, 69, large)-net in base 27, because
- 21 times m-reduction [i] would yield (46, 48, large)-net in base 27, but