Best Known (39, 39+19, s)-Nets in Base 25
(39, 39+19, 1737)-Net over F25 — Constructive and digital
Digital (39, 58, 1737)-net over F25, using
- 251 times duplication [i] based on digital (38, 57, 1737)-net over F25, using
- net defined by OOA [i] based on linear OOA(2557, 1737, F25, 19, 19) (dual of [(1737, 19), 32946, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2557, 15634, F25, 19) (dual of [15634, 15577, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2557, 15636, F25, 19) (dual of [15636, 15579, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(2555, 15625, F25, 19) (dual of [15625, 15570, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2546, 15625, F25, 16) (dual of [15625, 15579, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(252, 11, F25, 2) (dual of [11, 9, 3]-code or 11-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(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(2557, 15636, F25, 19) (dual of [15636, 15579, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2557, 15634, F25, 19) (dual of [15634, 15577, 20]-code), using
- net defined by OOA [i] based on linear OOA(2557, 1737, F25, 19, 19) (dual of [(1737, 19), 32946, 20]-NRT-code), using
(39, 39+19, 14545)-Net over F25 — Digital
Digital (39, 58, 14545)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2558, 14545, F25, 19) (dual of [14545, 14487, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2558, 15641, F25, 19) (dual of [15641, 15583, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(2555, 15626, F25, 19) (dual of [15626, 15571, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(2543, 15626, F25, 15) (dual of [15626, 15583, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(253, 15, F25, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,25) or 15-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 C([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2558, 15641, F25, 19) (dual of [15641, 15583, 20]-code), using
(39, 39+19, large)-Net in Base 25 — Upper bound on s
There is no (39, 58, large)-net in base 25, because
- 17 times m-reduction [i] would yield (39, 41, large)-net in base 25, but