Best Known (15, 15+14, s)-Nets in Base 81
(15, 15+14, 938)-Net over F81 — Constructive and digital
Digital (15, 29, 938)-net over F81, using
- 1 times m-reduction [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
(15, 15+14, 2963)-Net over F81 — Digital
Digital (15, 29, 2963)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8129, 2963, F81, 2, 14) (dual of [(2963, 2), 5897, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8129, 3284, F81, 2, 14) (dual of [(3284, 2), 6539, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8129, 6568, F81, 14) (dual of [6568, 6539, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(8129, 6569, F81, 14) (dual of [6569, 6540, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(8121, 6561, F81, 11) (dual of [6561, 6540, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(812, 8, F81, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(8129, 6569, F81, 14) (dual of [6569, 6540, 15]-code), using
- OOA 2-folding [i] based on linear OA(8129, 6568, F81, 14) (dual of [6568, 6539, 15]-code), using
- discarding factors / shortening the dual code based on linear OOA(8129, 3284, F81, 2, 14) (dual of [(3284, 2), 6539, 15]-NRT-code), using
(15, 15+14, 3407289)-Net in Base 81 — Upper bound on s
There is no (15, 29, 3407290)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 22 185326 153198 155765 893894 840047 719397 645888 206641 490401 > 8129 [i]