Best Known (56−19, 56, s)-Nets in Base 25
(56−19, 56, 1736)-Net over F25 — Constructive and digital
Digital (37, 56, 1736)-net over F25, using
- 251 times duplication [i] based on digital (36, 55, 1736)-net over F25, using
- net defined by OOA [i] based on linear OOA(2555, 1736, F25, 19, 19) (dual of [(1736, 19), 32929, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [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]
- OOA 9-folding and stacking with additional row [i] based on linear OA(2555, 15625, F25, 19) (dual of [15625, 15570, 20]-code), using
- net defined by OOA [i] based on linear OOA(2555, 1736, F25, 19, 19) (dual of [(1736, 19), 32929, 20]-NRT-code), using
(56−19, 56, 9957)-Net over F25 — Digital
Digital (37, 56, 9957)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2556, 9957, F25, 19) (dual of [9957, 9901, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2556, 15633, F25, 19) (dual of [15633, 15577, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [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(2549, 15626, F25, 17) (dual of [15626, 15577, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [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 C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2556, 15633, F25, 19) (dual of [15633, 15577, 20]-code), using
(56−19, 56, large)-Net in Base 25 — Upper bound on s
There is no (37, 56, large)-net in base 25, because
- 17 times m-reduction [i] would yield (37, 39, large)-net in base 25, but