Best Known (10, 10+9, s)-Nets in Base 25
(10, 10+9, 158)-Net over F25 — Constructive and digital
Digital (10, 19, 158)-net over F25, using
- net defined by OOA [i] based on linear OOA(2519, 158, F25, 9, 9) (dual of [(158, 9), 1403, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2519, 633, F25, 9) (dual of [633, 614, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(2517, 625, F25, 9) (dual of [625, 608, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2511, 625, F25, 6) (dual of [625, 614, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(252, 8, F25, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(2519, 633, F25, 9) (dual of [633, 614, 10]-code), using
(10, 10+9, 551)-Net over F25 — Digital
Digital (10, 19, 551)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2519, 551, F25, 9) (dual of [551, 532, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2519, 633, F25, 9) (dual of [633, 614, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(2517, 625, F25, 9) (dual of [625, 608, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2511, 625, F25, 6) (dual of [625, 614, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(252, 8, F25, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(2519, 633, F25, 9) (dual of [633, 614, 10]-code), using
(10, 10+9, 180122)-Net in Base 25 — Upper bound on s
There is no (10, 19, 180123)-net in base 25, because
- 1 times m-reduction [i] would yield (10, 18, 180123)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 14 552127 313025 531682 687265 > 2518 [i]