Best Known (54, 65, s)-Nets in Base 5
(54, 65, 78125)-Net over F5 — Constructive and digital
Digital (54, 65, 78125)-net over F5, using
- net defined by OOA [i] based on linear OOA(565, 78125, F5, 11, 11) (dual of [(78125, 11), 859310, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [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]
- OOA 5-folding and stacking with additional row [i] based on linear OA(565, 390626, F5, 11) (dual of [390626, 390561, 12]-code), using
(54, 65, 195313)-Net over F5 — Digital
Digital (54, 65, 195313)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(565, 195313, F5, 2, 11) (dual of [(195313, 2), 390561, 12]-NRT-code), using
- OOA 2-folding [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]
- OOA 2-folding [i] based on linear OA(565, 390626, F5, 11) (dual of [390626, 390561, 12]-code), using
(54, 65, large)-Net in Base 5 — Upper bound on s
There is no (54, 65, large)-net in base 5, because
- 9 times m-reduction [i] would yield (54, 56, large)-net in base 5, but