Best Known (98, 123, s)-Nets in Base 5
(98, 123, 1303)-Net over F5 — Constructive and digital
Digital (98, 123, 1303)-net over F5, using
- 51 times duplication [i] based on digital (97, 122, 1303)-net over F5, using
- net defined by OOA [i] based on linear OOA(5122, 1303, F5, 25, 25) (dual of [(1303, 25), 32453, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(5122, 15637, F5, 25) (dual of [15637, 15515, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5122, 15639, F5, 25) (dual of [15639, 15517, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(5121, 15626, F5, 25) (dual of [15626, 15505, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(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([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5122, 15639, F5, 25) (dual of [15639, 15517, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(5122, 15637, F5, 25) (dual of [15637, 15515, 26]-code), using
- net defined by OOA [i] based on linear OOA(5122, 1303, F5, 25, 25) (dual of [(1303, 25), 32453, 26]-NRT-code), using
(98, 123, 12006)-Net over F5 — Digital
Digital (98, 123, 12006)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5123, 12006, F5, 25) (dual of [12006, 11883, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5123, 15640, F5, 25) (dual of [15640, 15517, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5122, 15639, F5, 25) (dual of [15639, 15517, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(5121, 15626, F5, 25) (dual of [15626, 15505, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(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([0,12]) ⊂ C([0,11]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5122, 15639, F5, 25) (dual of [15639, 15517, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5123, 15640, F5, 25) (dual of [15640, 15517, 26]-code), using
(98, 123, large)-Net in Base 5 — Upper bound on s
There is no (98, 123, large)-net in base 5, because
- 23 times m-reduction [i] would yield (98, 100, large)-net in base 5, but