Best Known (21−7, 21, s)-Nets in Base 25
(21−7, 21, 5211)-Net over F25 — Constructive and digital
Digital (14, 21, 5211)-net over F25, using
- 251 times duplication [i] based on digital (13, 20, 5211)-net over F25, using
- net defined by OOA [i] based on linear OOA(2520, 5211, F25, 7, 7) (dual of [(5211, 7), 36457, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2520, 15634, F25, 7) (dual of [15634, 15614, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(2519, 15626, F25, 7) (dual of [15626, 15607, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(2513, 15626, F25, 5) (dual of [15626, 15613, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(257, 8, F25, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,25)), using
- dual of repetition code with length 8 [i]
- linear OA(251, 8, F25, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, 25, F25, 1) (dual of [25, 24, 2]-code), using
- Reed–Solomon code RS(24,25) [i]
- discarding factors / shortening the dual code based on linear OA(251, 25, F25, 1) (dual of [25, 24, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(2520, 15634, F25, 7) (dual of [15634, 15614, 8]-code), using
- net defined by OOA [i] based on linear OOA(2520, 5211, F25, 7, 7) (dual of [(5211, 7), 36457, 8]-NRT-code), using
(21−7, 21, 15636)-Net over F25 — Digital
Digital (14, 21, 15636)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2521, 15636, F25, 7) (dual of [15636, 15615, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(2519, 15625, F25, 7) (dual of [15625, 15606, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2510, 15625, F25, 4) (dual of [15625, 15615, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(252, 11, F25, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
(21−7, 21, large)-Net in Base 25 — Upper bound on s
There is no (14, 21, large)-net in base 25, because
- 5 times m-reduction [i] would yield (14, 16, large)-net in base 25, but