Best Known (31−11, 31, s)-Nets in Base 25
(31−11, 31, 3125)-Net over F25 — Constructive and digital
Digital (20, 31, 3125)-net over F25, using
- net defined by OOA [i] based on linear OOA(2531, 3125, F25, 11, 11) (dual of [(3125, 11), 34344, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2531, 15626, F25, 11) (dual of [15626, 15595, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, 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(2531, 15626, F25, 11) (dual of [15626, 15595, 12]-code), using
(31−11, 31, 7891)-Net over F25 — Digital
Digital (20, 31, 7891)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2531, 7891, F25, 11) (dual of [7891, 7860, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2531, 15625, F25, 11) (dual of [15625, 15594, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(2531, 15625, F25, 11) (dual of [15625, 15594, 12]-code), using
(31−11, 31, large)-Net in Base 25 — Upper bound on s
There is no (20, 31, large)-net in base 25, because
- 9 times m-reduction [i] would yield (20, 22, large)-net in base 25, but