Best Known (63−21, 63, s)-Nets in Base 25
(63−21, 63, 1563)-Net over F25 — Constructive and digital
Digital (42, 63, 1563)-net over F25, using
- 251 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(2562, 1563, F25, 21, 21) (dual of [(1563, 21), 32761, 22]-NRT-code), using
(63−21, 63, 12033)-Net over F25 — Digital
Digital (42, 63, 12033)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2563, 12033, F25, 21) (dual of [12033, 11970, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2563, 15636, F25, 21) (dual of [15636, 15573, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(2561, 15625, F25, 21) (dual of [15625, 15564, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2552, 15625, F25, 18) (dual of [15625, 15573, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(2563, 15636, F25, 21) (dual of [15636, 15573, 22]-code), using
(63−21, 63, large)-Net in Base 25 — Upper bound on s
There is no (42, 63, large)-net in base 25, because
- 19 times m-reduction [i] would yield (42, 44, large)-net in base 25, but