Best Known (55, 55+18, s)-Nets in Base 25
(55, 55+18, 43405)-Net over F25 — Constructive and digital
Digital (55, 73, 43405)-net over F25, using
- net defined by OOA [i] based on linear OOA(2573, 43405, F25, 18, 18) (dual of [(43405, 18), 781217, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2573, 390645, F25, 18) (dual of [390645, 390572, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2573, 390649, F25, 18) (dual of [390649, 390576, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [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(2549, 390625, F25, 13) (dual of [390625, 390576, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(254, 24, F25, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,25)), using
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- Reed–Solomon code RS(21,25) [i]
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(2573, 390649, F25, 18) (dual of [390649, 390576, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2573, 390645, F25, 18) (dual of [390645, 390572, 19]-code), using
(55, 55+18, 390649)-Net over F25 — Digital
Digital (55, 73, 390649)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2573, 390649, F25, 18) (dual of [390649, 390576, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [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(2549, 390625, F25, 13) (dual of [390625, 390576, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(254, 24, F25, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,25)), using
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- Reed–Solomon code RS(21,25) [i]
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
(55, 55+18, large)-Net in Base 25 — Upper bound on s
There is no (55, 73, large)-net in base 25, because
- 16 times m-reduction [i] would yield (55, 57, large)-net in base 25, but