Best Known (57, 57+20, s)-Nets in Base 25
(57, 57+20, 39062)-Net over F25 — Constructive and digital
Digital (57, 77, 39062)-net over F25, using
- net defined by OOA [i] based on linear OOA(2577, 39062, F25, 20, 20) (dual of [(39062, 20), 781163, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(2577, 390620, F25, 20) (dual of [390620, 390543, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2577, 390625, F25, 20) (dual of [390625, 390548, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2577, 390625, F25, 20) (dual of [390625, 390548, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(2577, 390620, F25, 20) (dual of [390620, 390543, 21]-code), using
(57, 57+20, 251372)-Net over F25 — Digital
Digital (57, 77, 251372)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2577, 251372, F25, 20) (dual of [251372, 251295, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2577, 390625, F25, 20) (dual of [390625, 390548, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2577, 390625, F25, 20) (dual of [390625, 390548, 21]-code), using
(57, 57+20, large)-Net in Base 25 — Upper bound on s
There is no (57, 77, large)-net in base 25, because
- 18 times m-reduction [i] would yield (57, 59, large)-net in base 25, but