Best Known (33, 33+15, s)-Nets in Base 27
(33, 33+15, 2815)-Net over F27 — Constructive and digital
Digital (33, 48, 2815)-net over F27, using
- net defined by OOA [i] based on linear OOA(2748, 2815, F27, 15, 15) (dual of [(2815, 15), 42177, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2748, 19706, F27, 15) (dual of [19706, 19658, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2748, 19707, F27, 15) (dual of [19707, 19659, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- linear OA(2743, 19684, F27, 15) (dual of [19684, 19641, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(2725, 19684, F27, 9) (dual of [19684, 19659, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(275, 23, F27, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,27)), using
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- Reed–Solomon code RS(22,27) [i]
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2748, 19707, F27, 15) (dual of [19707, 19659, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2748, 19706, F27, 15) (dual of [19706, 19658, 16]-code), using
(33, 33+15, 19707)-Net over F27 — Digital
Digital (33, 48, 19707)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2748, 19707, F27, 15) (dual of [19707, 19659, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- linear OA(2743, 19684, F27, 15) (dual of [19684, 19641, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(2725, 19684, F27, 9) (dual of [19684, 19659, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(275, 23, F27, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,27)), using
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- Reed–Solomon code RS(22,27) [i]
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
(33, 33+15, large)-Net in Base 27 — Upper bound on s
There is no (33, 48, large)-net in base 27, because
- 13 times m-reduction [i] would yield (33, 35, large)-net in base 27, but