Best Known (50, 50+25, s)-Nets in Base 25
(50, 50+25, 1302)-Net over F25 — Constructive and digital
Digital (50, 75, 1302)-net over F25, using
- 252 times duplication [i] based on digital (48, 73, 1302)-net over F25, using
- net defined by OOA [i] based on linear OOA(2573, 1302, F25, 25, 25) (dual of [(1302, 25), 32477, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2573, 15625, F25, 25) (dual of [15625, 15552, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2573, 15626, F25, 25) (dual of [15626, 15553, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2573, 15626, F25, 25) (dual of [15626, 15553, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2573, 15625, F25, 25) (dual of [15625, 15552, 26]-code), using
- net defined by OOA [i] based on linear OOA(2573, 1302, F25, 25, 25) (dual of [(1302, 25), 32477, 26]-NRT-code), using
(50, 50+25, 12347)-Net over F25 — Digital
Digital (50, 75, 12347)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2575, 12347, F25, 25) (dual of [12347, 12272, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2575, 15636, F25, 25) (dual of [15636, 15561, 26]-code), using
- 1 times truncation [i] based on linear OA(2576, 15637, F25, 26) (dual of [15637, 15561, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(2573, 15625, F25, 26) (dual of [15625, 15552, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2564, 15625, F25, 22) (dual of [15625, 15561, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(253, 12, F25, 3) (dual of [12, 9, 4]-code or 12-arc in PG(2,25) or 12-cap in PG(2,25)), using
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- Reed–Solomon code RS(22,25) [i]
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- 1 times truncation [i] based on linear OA(2576, 15637, F25, 26) (dual of [15637, 15561, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2575, 15636, F25, 25) (dual of [15636, 15561, 26]-code), using
(50, 50+25, large)-Net in Base 25 — Upper bound on s
There is no (50, 75, large)-net in base 25, because
- 23 times m-reduction [i] would yield (50, 52, large)-net in base 25, but