Best Known (57−15, 57, s)-Nets in Base 25
(57−15, 57, 55804)-Net over F25 — Constructive and digital
Digital (42, 57, 55804)-net over F25, using
- net defined by OOA [i] based on linear OOA(2557, 55804, F25, 15, 15) (dual of [(55804, 15), 837003, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2557, 390629, F25, 15) (dual of [390629, 390572, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(2557, 390625, F25, 15) (dual of [390625, 390568, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2553, 390625, F25, 14) (dual of [390625, 390572, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(250, 4, F25, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(2557, 390629, F25, 15) (dual of [390629, 390572, 16]-code), using
(57−15, 57, 248359)-Net over F25 — Digital
Digital (42, 57, 248359)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2557, 248359, F25, 15) (dual of [248359, 248302, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2557, 390625, F25, 15) (dual of [390625, 390568, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(2557, 390625, F25, 15) (dual of [390625, 390568, 16]-code), using
(57−15, 57, large)-Net in Base 25 — Upper bound on s
There is no (42, 57, large)-net in base 25, because
- 13 times m-reduction [i] would yield (42, 44, large)-net in base 25, but