Best Known (55, 55+30, s)-Nets in Base 25
(55, 55+30, 1041)-Net over F25 — Constructive and digital
Digital (55, 85, 1041)-net over F25, using
- net defined by OOA [i] based on linear OOA(2585, 1041, F25, 30, 30) (dual of [(1041, 30), 31145, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(2585, 15615, F25, 30) (dual of [15615, 15530, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2585, 15625, F25, 30) (dual of [15625, 15540, 31]-code), using
- an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- discarding factors / shortening the dual code based on linear OA(2585, 15625, F25, 30) (dual of [15625, 15540, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(2585, 15615, F25, 30) (dual of [15615, 15530, 31]-code), using
(55, 55+30, 7814)-Net over F25 — Digital
Digital (55, 85, 7814)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2585, 7814, F25, 2, 30) (dual of [(7814, 2), 15543, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2585, 15628, F25, 30) (dual of [15628, 15543, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(2585, 15625, F25, 30) (dual of [15625, 15540, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2582, 15625, F25, 29) (dual of [15625, 15543, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(250, 3, F25, 0) (dual of [3, 3, 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(29) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(2585, 15628, F25, 30) (dual of [15628, 15543, 31]-code), using
(55, 55+30, large)-Net in Base 25 — Upper bound on s
There is no (55, 85, large)-net in base 25, because
- 28 times m-reduction [i] would yield (55, 57, large)-net in base 25, but