Best Known (58, 58+19, s)-Nets in Base 25
(58, 58+19, 43405)-Net over F25 — Constructive and digital
Digital (58, 77, 43405)-net over F25, using
- net defined by OOA [i] based on linear OOA(2577, 43405, F25, 19, 19) (dual of [(43405, 19), 824618, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2577, 390646, F25, 19) (dual of [390646, 390569, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2577, 390649, F25, 19) (dual of [390649, 390572, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(2573, 390625, F25, 19) (dual of [390625, 390552, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(254, 24, F25, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,25)), using
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- Reed–Solomon code RS(21,25) [i]
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(2577, 390649, F25, 19) (dual of [390649, 390572, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2577, 390646, F25, 19) (dual of [390646, 390569, 20]-code), using
(58, 58+19, 390649)-Net over F25 — Digital
Digital (58, 77, 390649)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2577, 390649, F25, 19) (dual of [390649, 390572, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(2573, 390625, F25, 19) (dual of [390625, 390552, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(254, 24, F25, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,25)), using
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- Reed–Solomon code RS(21,25) [i]
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
(58, 58+19, large)-Net in Base 25 — Upper bound on s
There is no (58, 77, large)-net in base 25, because
- 17 times m-reduction [i] would yield (58, 60, large)-net in base 25, but