Best Known (62−21, 62, s)-Nets in Base 25
(62−21, 62, 1563)-Net over F25 — Constructive and digital
Digital (41, 62, 1563)-net over F25, using
- net defined by OOA [i] based on linear OOA(2562, 1563, F25, 21, 21) (dual of [(1563, 21), 32761, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2562, 15631, F25, 21) (dual of [15631, 15569, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2562, 15633, F25, 21) (dual of [15633, 15571, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(2561, 15626, F25, 21) (dual of [15626, 15565, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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(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 C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2562, 15633, F25, 21) (dual of [15633, 15571, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2562, 15631, F25, 21) (dual of [15631, 15569, 22]-code), using
(62−21, 62, 10156)-Net over F25 — Digital
Digital (41, 62, 10156)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2562, 10156, F25, 21) (dual of [10156, 10094, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2562, 15633, F25, 21) (dual of [15633, 15571, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(2561, 15626, F25, 21) (dual of [15626, 15565, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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(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 C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2562, 15633, F25, 21) (dual of [15633, 15571, 22]-code), using
(62−21, 62, large)-Net in Base 25 — Upper bound on s
There is no (41, 62, large)-net in base 25, because
- 19 times m-reduction [i] would yield (41, 43, large)-net in base 25, but