Best Known (22, 22+5, s)-Nets in Base 5
(22, 22+5, 15628)-Net over F5 — Constructive and digital
Digital (22, 27, 15628)-net over F5, using
- net defined by OOA [i] based on linear OOA(527, 15628, F5, 5, 5) (dual of [(15628, 5), 78113, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(527, 31257, F5, 5) (dual of [31257, 31230, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(526, 31256, F5, 5) (dual of [31256, 31230, 6]-code), using
- trace code [i] based on linear OA(2513, 15628, F25, 5) (dual of [15628, 15615, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(2513, 15625, F25, 5) (dual of [15625, 15612, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2510, 15625, F25, 4) (dual of [15625, 15615, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(250, 3, F25, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(2513, 15628, F25, 5) (dual of [15628, 15615, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(526, 31256, F5, 5) (dual of [31256, 31230, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(527, 31257, F5, 5) (dual of [31257, 31230, 6]-code), using
(22, 22+5, 31258)-Net over F5 — Digital
Digital (22, 27, 31258)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(527, 31258, F5, 5) (dual of [31258, 31231, 6]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(526, 31256, F5, 5) (dual of [31256, 31230, 6]-code), using
- trace code [i] based on linear OA(2513, 15628, F25, 5) (dual of [15628, 15615, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(2513, 15625, F25, 5) (dual of [15625, 15612, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2510, 15625, F25, 4) (dual of [15625, 15615, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(250, 3, F25, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(2513, 15628, F25, 5) (dual of [15628, 15615, 6]-code), using
- linear OA(526, 31257, F5, 4) (dual of [31257, 31231, 5]-code), using Gilbert–Varšamov bound and bm = 526 > Vbs−1(k−1) = 325 684923 376545 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(526, 31256, F5, 5) (dual of [31256, 31230, 6]-code), using
- construction X with Varšamov bound [i] based on
(22, 22+5, large)-Net in Base 5 — Upper bound on s
There is no (22, 27, large)-net in base 5, because
- 3 times m-reduction [i] would yield (22, 24, large)-net in base 5, but