Best Known (30, 30+11, s)-Nets in Base 25
(30, 30+11, 78125)-Net over F25 — Constructive and digital
Digital (30, 41, 78125)-net over F25, using
- net defined by OOA [i] based on linear OOA(2541, 78125, F25, 11, 11) (dual of [(78125, 11), 859334, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2541, 390626, F25, 11) (dual of [390626, 390585, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, 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(2541, 390626, F25, 11) (dual of [390626, 390585, 12]-code), using
(30, 30+11, 282229)-Net over F25 — Digital
Digital (30, 41, 282229)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2541, 282229, F25, 11) (dual of [282229, 282188, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2541, 390625, F25, 11) (dual of [390625, 390584, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(2541, 390625, F25, 11) (dual of [390625, 390584, 12]-code), using
(30, 30+11, large)-Net in Base 25 — Upper bound on s
There is no (30, 41, large)-net in base 25, because
- 9 times m-reduction [i] would yield (30, 32, large)-net in base 25, but