Best Known (54, 54+10, s)-Nets in Base 5
(54, 54+10, 78125)-Net over F5 — Constructive and digital
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
(54, 54+10, 300616)-Net over F5 — Digital
Digital (54, 64, 300616)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(564, 300616, F5, 10) (dual of [300616, 300552, 11]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(564, 390625, F5, 10) (dual of [390625, 390561, 11]-code), using
(54, 54+10, large)-Net in Base 5 — Upper bound on s
There is no (54, 64, large)-net in base 5, because
- 8 times m-reduction [i] would yield (54, 56, large)-net in base 5, but