Best Known (17, 17+17, s)-Nets in Base 81
(17, 17+17, 820)-Net over F81 — Constructive and digital
Digital (17, 34, 820)-net over F81, using
- 811 times duplication [i] based on digital (16, 33, 820)-net over F81, using
- net defined by OOA [i] based on linear OOA(8133, 820, F81, 17, 17) (dual of [(820, 17), 13907, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8133, 6561, F81, 17) (dual of [6561, 6528, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(8133, 6561, F81, 17) (dual of [6561, 6528, 18]-code), using
- net defined by OOA [i] based on linear OOA(8133, 820, F81, 17, 17) (dual of [(820, 17), 13907, 18]-NRT-code), using
(17, 17+17, 2189)-Net over F81 — Digital
Digital (17, 34, 2189)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8134, 2189, F81, 3, 17) (dual of [(2189, 3), 6533, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8134, 6567, F81, 17) (dual of [6567, 6533, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(8133, 6562, F81, 17) (dual of [6562, 6529, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 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(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([0,8]) ⊂ C([0,7]) [i] based on
- OOA 3-folding [i] based on linear OA(8134, 6567, F81, 17) (dual of [6567, 6533, 18]-code), using
(17, 17+17, 3508329)-Net in Base 81 — Upper bound on s
There is no (17, 34, 3508330)-net in base 81, because
- 1 times m-reduction [i] would yield (17, 33, 3508330)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 955 006736 128917 582080 999993 292259 510097 681083 723177 042514 259201 > 8133 [i]