Best Known (20−7, 20, s)-Nets in Base 27
(20−7, 20, 6563)-Net over F27 — Constructive and digital
Digital (13, 20, 6563)-net over F27, using
- net defined by OOA [i] based on linear OOA(2720, 6563, F27, 7, 7) (dual of [(6563, 7), 45921, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2720, 19690, F27, 7) (dual of [19690, 19670, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2720, 19691, F27, 7) (dual of [19691, 19671, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(2719, 19684, F27, 7) (dual of [19684, 19665, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(2713, 19684, F27, 5) (dual of [19684, 19671, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (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([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2720, 19691, F27, 7) (dual of [19691, 19671, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2720, 19690, F27, 7) (dual of [19690, 19670, 8]-code), using
(20−7, 20, 19692)-Net over F27 — Digital
Digital (13, 20, 19692)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2720, 19692, F27, 7) (dual of [19692, 19672, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(2719, 19684, F27, 7) (dual of [19684, 19665, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(2713, 19684, F27, 5) (dual of [19684, 19671, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(277, 8, F27, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,27)), using
- dual of repetition code with length 8 [i]
- linear OA(271, 8, F27, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, 27, F27, 1) (dual of [27, 26, 2]-code), using
- Reed–Solomon code RS(26,27) [i]
- discarding factors / shortening the dual code based on linear OA(271, 27, F27, 1) (dual of [27, 26, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
(20−7, 20, large)-Net in Base 27 — Upper bound on s
There is no (13, 20, large)-net in base 27, because
- 5 times m-reduction [i] would yield (13, 15, large)-net in base 27, but