Best Known (67−22, 67, s)-Nets in Base 27
(67−22, 67, 1790)-Net over F27 — Constructive and digital
Digital (45, 67, 1790)-net over F27, using
- 1 times m-reduction [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
(67−22, 67, 16888)-Net over F27 — Digital
Digital (45, 67, 16888)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2767, 16888, F27, 22) (dual of [16888, 16821, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2767, 19698, F27, 22) (dual of [19698, 19631, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(2764, 19683, F27, 22) (dual of [19683, 19619, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2752, 19683, F27, 18) (dual of [19683, 19631, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(273, 15, F27, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,27) or 15-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(2767, 19698, F27, 22) (dual of [19698, 19631, 23]-code), using
(67−22, 67, large)-Net in Base 27 — Upper bound on s
There is no (45, 67, large)-net in base 27, because
- 20 times m-reduction [i] would yield (45, 47, large)-net in base 27, but