Best Known (40, 40+13, s)-Nets in Base 25
(40, 40+13, 65108)-Net over F25 — Constructive and digital
Digital (40, 53, 65108)-net over F25, using
- net defined by OOA [i] based on linear OOA(2553, 65108, F25, 13, 13) (dual of [(65108, 13), 846351, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2553, 390649, F25, 13) (dual of [390649, 390596, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(2549, 390625, F25, 13) (dual of [390625, 390576, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2529, 390625, F25, 8) (dual of [390625, 390596, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(12) ⊂ Ce(7) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(2553, 390649, F25, 13) (dual of [390649, 390596, 14]-code), using
(40, 40+13, 390649)-Net over F25 — Digital
Digital (40, 53, 390649)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2553, 390649, F25, 13) (dual of [390649, 390596, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(2549, 390625, F25, 13) (dual of [390625, 390576, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2529, 390625, F25, 8) (dual of [390625, 390596, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(12) ⊂ Ce(7) [i] based on
(40, 40+13, large)-Net in Base 25 — Upper bound on s
There is no (40, 53, large)-net in base 25, because
- 11 times m-reduction [i] would yield (40, 42, large)-net in base 25, but