Best Known (27, 27+14, s)-Nets in Base 25
(27, 27+14, 2233)-Net over F25 — Constructive and digital
Digital (27, 41, 2233)-net over F25, using
- net defined by OOA [i] based on linear OOA(2541, 2233, F25, 14, 14) (dual of [(2233, 14), 31221, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2541, 15631, F25, 14) (dual of [15631, 15590, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2541, 15632, F25, 14) (dual of [15632, 15591, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [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(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(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(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(2541, 15632, F25, 14) (dual of [15632, 15591, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(2541, 15631, F25, 14) (dual of [15631, 15590, 15]-code), using
(27, 27+14, 10063)-Net over F25 — Digital
Digital (27, 41, 10063)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2541, 10063, F25, 14) (dual of [10063, 10022, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2541, 15632, F25, 14) (dual of [15632, 15591, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [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(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(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(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(2541, 15632, F25, 14) (dual of [15632, 15591, 15]-code), using
(27, 27+14, large)-Net in Base 25 — Upper bound on s
There is no (27, 41, large)-net in base 25, because
- 12 times m-reduction [i] would yield (27, 29, large)-net in base 25, but