Best Known (52, 70, s)-Nets in Base 25
(52, 70, 43403)-Net over F25 — Constructive and digital
Digital (52, 70, 43403)-net over F25, using
- 251 times duplication [i] based on digital (51, 69, 43403)-net over F25, using
- net defined by OOA [i] based on linear OOA(2569, 43403, F25, 18, 18) (dual of [(43403, 18), 781185, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2569, 390627, F25, 18) (dual of [390627, 390558, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2569, 390629, F25, 18) (dual of [390629, 390560, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(2569, 390625, F25, 18) (dual of [390625, 390556, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2565, 390625, F25, 17) (dual of [390625, 390560, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(250, 4, F25, 0) (dual of [4, 4, 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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(2569, 390629, F25, 18) (dual of [390629, 390560, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2569, 390627, F25, 18) (dual of [390627, 390558, 19]-code), using
- net defined by OOA [i] based on linear OOA(2569, 43403, F25, 18, 18) (dual of [(43403, 18), 781185, 19]-NRT-code), using
(52, 70, 302644)-Net over F25 — Digital
Digital (52, 70, 302644)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2570, 302644, F25, 18) (dual of [302644, 302574, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2570, 390634, F25, 18) (dual of [390634, 390564, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(2569, 390625, F25, 18) (dual of [390625, 390556, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2561, 390625, F25, 16) (dual of [390625, 390564, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(251, 9, F25, 1) (dual of [9, 8, 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(2570, 390634, F25, 18) (dual of [390634, 390564, 19]-code), using
(52, 70, large)-Net in Base 25 — Upper bound on s
There is no (52, 70, large)-net in base 25, because
- 16 times m-reduction [i] would yield (52, 54, large)-net in base 25, but