Best Known (39, 39+20, s)-Nets in Base 25
(39, 39+20, 1563)-Net over F25 — Constructive and digital
Digital (39, 59, 1563)-net over F25, using
- net defined by OOA [i] based on linear OOA(2559, 1563, F25, 20, 20) (dual of [(1563, 20), 31201, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(2559, 15630, F25, 20) (dual of [15630, 15571, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2559, 15632, F25, 20) (dual of [15632, 15573, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(2558, 15625, F25, 20) (dual of [15625, 15567, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2552, 15625, F25, 18) (dual of [15625, 15573, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(251, 7, F25, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(2559, 15632, F25, 20) (dual of [15632, 15573, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(2559, 15630, F25, 20) (dual of [15630, 15571, 21]-code), using
(39, 39+20, 10047)-Net over F25 — Digital
Digital (39, 59, 10047)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2559, 10047, F25, 20) (dual of [10047, 9988, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2559, 15632, F25, 20) (dual of [15632, 15573, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(2558, 15625, F25, 20) (dual of [15625, 15567, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2552, 15625, F25, 18) (dual of [15625, 15573, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(251, 7, F25, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(2559, 15632, F25, 20) (dual of [15632, 15573, 21]-code), using
(39, 39+20, large)-Net in Base 25 — Upper bound on s
There is no (39, 59, large)-net in base 25, because
- 18 times m-reduction [i] would yield (39, 41, large)-net in base 25, but