Best Known (26, 40, s)-Nets in Base 25
(26, 40, 2232)-Net over F25 — Constructive and digital
Digital (26, 40, 2232)-net over F25, using
- net defined by OOA [i] based on linear OOA(2540, 2232, F25, 14, 14) (dual of [(2232, 14), 31208, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2540, 15624, F25, 14) (dual of [15624, 15584, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2540, 15625, F25, 14) (dual of [15625, 15585, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(2540, 15625, F25, 14) (dual of [15625, 15585, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(2540, 15624, F25, 14) (dual of [15624, 15584, 15]-code), using
(26, 40, 7814)-Net over F25 — Digital
Digital (26, 40, 7814)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2540, 7814, F25, 2, 14) (dual of [(7814, 2), 15588, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2540, 15628, F25, 14) (dual of [15628, 15588, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(2540, 15625, F25, 14) (dual of [15625, 15585, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2537, 15625, F25, 13) (dual of [15625, 15588, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(250, 3, F25, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(2540, 15628, F25, 14) (dual of [15628, 15588, 15]-code), using
(26, 40, large)-Net in Base 25 — Upper bound on s
There is no (26, 40, large)-net in base 25, because
- 12 times m-reduction [i] would yield (26, 28, large)-net in base 25, but