Best Known (78−25, 78, s)-Nets in Base 25
(78−25, 78, 1304)-Net over F25 — Constructive and digital
Digital (53, 78, 1304)-net over F25, using
- net defined by OOA [i] based on linear OOA(2578, 1304, F25, 25, 25) (dual of [(1304, 25), 32522, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2578, 15649, F25, 25) (dual of [15649, 15571, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(2573, 15626, F25, 25) (dual of [15626, 15553, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2555, 15626, F25, 19) (dual of [15626, 15571, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(255, 23, F25, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,25)), using
- discarding factors / shortening the dual code based on linear OA(255, 25, F25, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
- Reed–Solomon code RS(20,25) [i]
- discarding factors / shortening the dual code based on linear OA(255, 25, F25, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(2578, 15649, F25, 25) (dual of [15649, 15571, 26]-code), using
(78−25, 78, 15649)-Net over F25 — Digital
Digital (53, 78, 15649)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2578, 15649, F25, 25) (dual of [15649, 15571, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(2573, 15626, F25, 25) (dual of [15626, 15553, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2555, 15626, F25, 19) (dual of [15626, 15571, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(255, 23, F25, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,25)), using
- discarding factors / shortening the dual code based on linear OA(255, 25, F25, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
- Reed–Solomon code RS(20,25) [i]
- discarding factors / shortening the dual code based on linear OA(255, 25, F25, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
(78−25, 78, large)-Net in Base 25 — Upper bound on s
There is no (53, 78, large)-net in base 25, because
- 23 times m-reduction [i] would yield (53, 55, large)-net in base 25, but