Best Known (71−10, 71, s)-Nets in Base 5
(71−10, 71, 78146)-Net over F5 — Constructive and digital
Digital (61, 71, 78146)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 21)-net over F5, using
- digital (54, 64, 78125)-net over F5, using
- net defined by OOA [i] based on linear OOA(564, 78125, F5, 10, 10) (dual of [(78125, 10), 781186, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(564, 390625, F5, 10) (dual of [390625, 390561, 11]-code), using
- 1 times truncation [i] based on linear OA(565, 390626, F5, 11) (dual of [390626, 390561, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(565, 390626, F5, 11) (dual of [390626, 390561, 12]-code), using
- OA 5-folding and stacking [i] based on linear OA(564, 390625, F5, 10) (dual of [390625, 390561, 11]-code), using
- net defined by OOA [i] based on linear OOA(564, 78125, F5, 10, 10) (dual of [(78125, 10), 781186, 11]-NRT-code), using
(71−10, 71, 390664)-Net over F5 — Digital
Digital (61, 71, 390664)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(571, 390664, F5, 10) (dual of [390664, 390593, 11]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(570, 390662, F5, 10) (dual of [390662, 390592, 11]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
- linear OA(565, 390625, F5, 11) (dual of [390625, 390560, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(533, 390625, F5, 6) (dual of [390625, 390592, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(55, 37, F5, 3) (dual of [37, 32, 4]-code or 37-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
- linear OA(570, 390663, F5, 9) (dual of [390663, 390593, 10]-code), using Gilbert–Varšamov bound and bm = 570 > Vbs−1(k−1) = 881 738605 387522 443526 611262 395026 581465 113865 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(570, 390662, F5, 10) (dual of [390662, 390592, 11]-code), using
- construction X with Varšamov bound [i] based on
(71−10, 71, large)-Net in Base 5 — Upper bound on s
There is no (61, 71, large)-net in base 5, because
- 8 times m-reduction [i] would yield (61, 63, large)-net in base 5, but