Best Known (112−23, 112, s)-Nets in Base 5
(112−23, 112, 1421)-Net over F5 — Constructive and digital
Digital (89, 112, 1421)-net over F5, using
- 52 times duplication [i] based on digital (87, 110, 1421)-net over F5, using
- net defined by OOA [i] based on linear OOA(5110, 1421, F5, 23, 23) (dual of [(1421, 23), 32573, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5110, 15632, F5, 23) (dual of [15632, 15522, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5110, 15639, F5, 23) (dual of [15639, 15529, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(5109, 15626, F5, 23) (dual of [15626, 15517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(597, 15626, F5, 21) (dual of [15626, 15529, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(51, 13, F5, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 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(5110, 15639, F5, 23) (dual of [15639, 15529, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5110, 15632, F5, 23) (dual of [15632, 15522, 24]-code), using
- net defined by OOA [i] based on linear OOA(5110, 1421, F5, 23, 23) (dual of [(1421, 23), 32573, 24]-NRT-code), using
(112−23, 112, 10725)-Net over F5 — Digital
Digital (89, 112, 10725)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5112, 10725, F5, 23) (dual of [10725, 10613, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5112, 15641, F5, 23) (dual of [15641, 15529, 24]-code), using
- 2 times code embedding in larger space [i] based on linear OA(5110, 15639, F5, 23) (dual of [15639, 15529, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(5109, 15626, F5, 23) (dual of [15626, 15517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(597, 15626, F5, 21) (dual of [15626, 15529, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(51, 13, F5, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 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
- 2 times code embedding in larger space [i] based on linear OA(5110, 15639, F5, 23) (dual of [15639, 15529, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5112, 15641, F5, 23) (dual of [15641, 15529, 24]-code), using
(112−23, 112, large)-Net in Base 5 — Upper bound on s
There is no (89, 112, large)-net in base 5, because
- 21 times m-reduction [i] would yield (89, 91, large)-net in base 5, but