Best Known (35, 35+18, s)-Nets in Base 25
(35, 35+18, 1736)-Net over F25 — Constructive and digital
Digital (35, 53, 1736)-net over F25, using
- 251 times duplication [i] based on digital (34, 52, 1736)-net over F25, using
- net defined by OOA [i] based on linear OOA(2552, 1736, F25, 18, 18) (dual of [(1736, 18), 31196, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2552, 15624, F25, 18) (dual of [15624, 15572, 19]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(2552, 15625, F25, 18) (dual of [15625, 15573, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2552, 15624, F25, 18) (dual of [15624, 15572, 19]-code), using
- net defined by OOA [i] based on linear OOA(2552, 1736, F25, 18, 18) (dual of [(1736, 18), 31196, 19]-NRT-code), using
(35, 35+18, 9892)-Net over F25 — Digital
Digital (35, 53, 9892)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2553, 9892, F25, 18) (dual of [9892, 9839, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2553, 15632, F25, 18) (dual of [15632, 15579, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- 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(2546, 15625, F25, 16) (dual of [15625, 15579, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(17) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(2553, 15632, F25, 18) (dual of [15632, 15579, 19]-code), using
(35, 35+18, large)-Net in Base 25 — Upper bound on s
There is no (35, 53, large)-net in base 25, because
- 16 times m-reduction [i] would yield (35, 37, large)-net in base 25, but