Best Known (68, 104, s)-Nets in Base 25
(68, 104, 868)-Net over F25 — Constructive and digital
Digital (68, 104, 868)-net over F25, using
- 251 times duplication [i] based on digital (67, 103, 868)-net over F25, using
- net defined by OOA [i] based on linear OOA(25103, 868, F25, 36, 36) (dual of [(868, 36), 31145, 37]-NRT-code), using
- OA 18-folding and stacking [i] based on linear OA(25103, 15624, F25, 36) (dual of [15624, 15521, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(25103, 15625, F25, 36) (dual of [15625, 15522, 37]-code), using
- an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- discarding factors / shortening the dual code based on linear OA(25103, 15625, F25, 36) (dual of [15625, 15522, 37]-code), using
- OA 18-folding and stacking [i] based on linear OA(25103, 15624, F25, 36) (dual of [15624, 15521, 37]-code), using
- net defined by OOA [i] based on linear OOA(25103, 868, F25, 36, 36) (dual of [(868, 36), 31145, 37]-NRT-code), using
(68, 104, 9670)-Net over F25 — Digital
Digital (68, 104, 9670)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25104, 9670, F25, 36) (dual of [9670, 9566, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(25104, 15632, F25, 36) (dual of [15632, 15528, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(33) [i] based on
- linear OA(25103, 15625, F25, 36) (dual of [15625, 15522, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2597, 15625, F25, 34) (dual of [15625, 15528, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [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(35) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(25104, 15632, F25, 36) (dual of [15632, 15528, 37]-code), using
(68, 104, large)-Net in Base 25 — Upper bound on s
There is no (68, 104, large)-net in base 25, because
- 34 times m-reduction [i] would yield (68, 70, large)-net in base 25, but