Best Known (122−19, 122, s)-Nets in Base 5
(122−19, 122, 43404)-Net over F5 — Constructive and digital
Digital (103, 122, 43404)-net over F5, using
- net defined by OOA [i] based on linear OOA(5122, 43404, F5, 19, 19) (dual of [(43404, 19), 824554, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5122, 390637, F5, 19) (dual of [390637, 390515, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5122, 390642, F5, 19) (dual of [390642, 390520, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(5105, 390625, F5, 17) (dual of [390625, 390520, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(51, 17, F5, 1) (dual of [17, 16, 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(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(5122, 390642, F5, 19) (dual of [390642, 390520, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5122, 390637, F5, 19) (dual of [390637, 390515, 20]-code), using
(122−19, 122, 195321)-Net over F5 — Digital
Digital (103, 122, 195321)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5122, 195321, F5, 2, 19) (dual of [(195321, 2), 390520, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5122, 390642, F5, 19) (dual of [390642, 390520, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(5121, 390625, F5, 19) (dual of [390625, 390504, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(5105, 390625, F5, 17) (dual of [390625, 390520, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(51, 17, F5, 1) (dual of [17, 16, 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(18) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(5122, 390642, F5, 19) (dual of [390642, 390520, 20]-code), using
(122−19, 122, large)-Net in Base 5 — Upper bound on s
There is no (103, 122, large)-net in base 5, because
- 17 times m-reduction [i] would yield (103, 105, large)-net in base 5, but