Best Known (42, 42+21, s)-Nets in Base 27
(42, 42+21, 1969)-Net over F27 — Constructive and digital
Digital (42, 63, 1969)-net over F27, using
- 271 times duplication [i] based on digital (41, 62, 1969)-net over F27, using
- net defined by OOA [i] based on linear OOA(2762, 1969, F27, 21, 21) (dual of [(1969, 21), 41287, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2762, 19691, F27, 21) (dual of [19691, 19629, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- 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(2755, 19684, F27, 19) (dual of [19684, 19629, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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([0,10]) ⊂ C([0,9]) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(2762, 19691, F27, 21) (dual of [19691, 19629, 22]-code), using
- net defined by OOA [i] based on linear OOA(2762, 1969, F27, 21, 21) (dual of [(1969, 21), 41287, 22]-NRT-code), using
(42, 42+21, 14280)-Net over F27 — Digital
Digital (42, 63, 14280)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2763, 14280, F27, 21) (dual of [14280, 14217, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2763, 19694, F27, 21) (dual of [19694, 19631, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(2761, 19683, F27, 21) (dual of [19683, 19622, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(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(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(2763, 19694, F27, 21) (dual of [19694, 19631, 22]-code), using
(42, 42+21, large)-Net in Base 27 — Upper bound on s
There is no (42, 63, large)-net in base 27, because
- 19 times m-reduction [i] would yield (42, 44, large)-net in base 27, but