Best Known (116−24, 116, s)-Nets in Base 5
(116−24, 116, 1303)-Net over F5 — Constructive and digital
Digital (92, 116, 1303)-net over F5, using
- net defined by OOA [i] based on linear OOA(5116, 1303, F5, 24, 24) (dual of [(1303, 24), 31156, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(5116, 15636, F5, 24) (dual of [15636, 15520, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(5116, 15638, F5, 24) (dual of [15638, 15522, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(5115, 15625, F5, 24) (dual of [15625, 15510, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(5103, 15625, F5, 22) (dual of [15625, 15522, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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 Ce(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(5116, 15638, F5, 24) (dual of [15638, 15522, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(5116, 15636, F5, 24) (dual of [15636, 15520, 25]-code), using
(116−24, 116, 10182)-Net over F5 — Digital
Digital (92, 116, 10182)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5116, 10182, F5, 24) (dual of [10182, 10066, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(5116, 15638, F5, 24) (dual of [15638, 15522, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(5115, 15625, F5, 24) (dual of [15625, 15510, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(5103, 15625, F5, 22) (dual of [15625, 15522, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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 Ce(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(5116, 15638, F5, 24) (dual of [15638, 15522, 25]-code), using
(116−24, 116, 7551113)-Net in Base 5 — Upper bound on s
There is no (92, 116, 7551114)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1203 707928 676959 657990 771448 226209 324718 858761 447527 432358 788410 137309 917576 531393 > 5116 [i]