Best Known (45, 45+23, s)-Nets in Base 25
(45, 45+23, 1421)-Net over F25 — Constructive and digital
Digital (45, 68, 1421)-net over F25, using
- net defined by OOA [i] based on linear OOA(2568, 1421, F25, 23, 23) (dual of [(1421, 23), 32615, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2568, 15632, F25, 23) (dual of [15632, 15564, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2568, 15633, F25, 23) (dual of [15633, 15565, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(2567, 15626, F25, 23) (dual of [15626, 15559, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- 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(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([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2568, 15633, F25, 23) (dual of [15633, 15565, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2568, 15632, F25, 23) (dual of [15632, 15564, 24]-code), using
(45, 45+23, 10422)-Net over F25 — Digital
Digital (45, 68, 10422)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2568, 10422, F25, 23) (dual of [10422, 10354, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2568, 15633, F25, 23) (dual of [15633, 15565, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(2567, 15626, F25, 23) (dual of [15626, 15559, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- 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(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([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2568, 15633, F25, 23) (dual of [15633, 15565, 24]-code), using
(45, 45+23, large)-Net in Base 25 — Upper bound on s
There is no (45, 68, large)-net in base 25, because
- 21 times m-reduction [i] would yield (45, 47, large)-net in base 25, but