Best Known (64−21, 64, s)-Nets in Base 27
(64−21, 64, 1969)-Net over F27 — Constructive and digital
Digital (43, 64, 1969)-net over F27, using
- 272 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
(64−21, 64, 16987)-Net over F27 — Digital
Digital (43, 64, 16987)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2764, 16987, F27, 21) (dual of [16987, 16923, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2764, 19699, F27, 21) (dual of [19699, 19635, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [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(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(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 C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2764, 19699, F27, 21) (dual of [19699, 19635, 22]-code), using
(64−21, 64, large)-Net in Base 27 — Upper bound on s
There is no (43, 64, large)-net in base 27, because
- 19 times m-reduction [i] would yield (43, 45, large)-net in base 27, but