Best Known (34−12, 34, s)-Nets in Base 25
(34−12, 34, 2604)-Net over F25 — Constructive and digital
Digital (22, 34, 2604)-net over F25, using
- net defined by OOA [i] based on linear OOA(2534, 2604, F25, 12, 12) (dual of [(2604, 12), 31214, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2534, 15624, F25, 12) (dual of [15624, 15590, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2534, 15625, F25, 12) (dual of [15625, 15591, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(2534, 15625, F25, 12) (dual of [15625, 15591, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(2534, 15624, F25, 12) (dual of [15624, 15590, 13]-code), using
(34−12, 34, 7814)-Net over F25 — Digital
Digital (22, 34, 7814)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2534, 7814, F25, 2, 12) (dual of [(7814, 2), 15594, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2534, 15628, F25, 12) (dual of [15628, 15594, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(2534, 15625, F25, 12) (dual of [15625, 15591, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2531, 15625, F25, 11) (dual of [15625, 15594, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(2534, 15628, F25, 12) (dual of [15628, 15594, 13]-code), using
(34−12, 34, large)-Net in Base 25 — Upper bound on s
There is no (22, 34, large)-net in base 25, because
- 10 times m-reduction [i] would yield (22, 24, large)-net in base 25, but