Best Known (17, 17+15, s)-Nets in Base 81
(17, 17+15, 938)-Net over F81 — Constructive and digital
Digital (17, 32, 938)-net over F81, using
- 812 times duplication [i] based on digital (15, 30, 938)-net over F81, using
- net defined by OOA [i] based on linear OOA(8130, 938, F81, 15, 15) (dual of [(938, 15), 14040, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8130, 6567, F81, 15) (dual of [6567, 6537, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(8129, 6562, F81, 15) (dual of [6562, 6533, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(8125, 6562, F81, 13) (dual of [6562, 6537, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(8130, 6567, F81, 15) (dual of [6567, 6537, 16]-code), using
- net defined by OOA [i] based on linear OOA(8130, 938, F81, 15, 15) (dual of [(938, 15), 14040, 16]-NRT-code), using
(17, 17+15, 3286)-Net over F81 — Digital
Digital (17, 32, 3286)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8132, 3286, F81, 2, 15) (dual of [(3286, 2), 6540, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8132, 6572, F81, 15) (dual of [6572, 6540, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8132, 6573, F81, 15) (dual of [6573, 6541, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(8129, 6562, F81, 15) (dual of [6562, 6533, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(8121, 6562, F81, 11) (dual of [6562, 6541, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(813, 11, F81, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8132, 6573, F81, 15) (dual of [6573, 6541, 16]-code), using
- OOA 2-folding [i] based on linear OA(8132, 6572, F81, 15) (dual of [6572, 6540, 16]-code), using
(17, 17+15, large)-Net in Base 81 — Upper bound on s
There is no (17, 32, large)-net in base 81, because
- 13 times m-reduction [i] would yield (17, 19, large)-net in base 81, but