Best Known (26−4, 26, s)-Nets in Base 5
(26−4, 26, 390629)-Net over F5 — Constructive and digital
Digital (22, 26, 390629)-net over F5, using
- net defined by OOA [i] based on linear OOA(526, 390629, F5, 4, 4) (dual of [(390629, 4), 1562490, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(526, 390629, F5, 3, 4) (dual of [(390629, 3), 1171861, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(526, 781258, F5, 4) (dual of [781258, 781232, 5]-code), using
- trace code [i] based on linear OA(2513, 390629, F25, 4) (dual of [390629, 390616, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(2513, 390625, F25, 4) (dual of [390625, 390612, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(259, 390625, F25, 3) (dual of [390625, 390616, 4]-code or 390625-cap in PG(8,25)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(250, 4, F25, 0) (dual of [4, 4, 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(3) ⊂ Ce(2) [i] based on
- trace code [i] based on linear OA(2513, 390629, F25, 4) (dual of [390629, 390616, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(526, 781258, F5, 4) (dual of [781258, 781232, 5]-code), using
- appending kth column [i] based on linear OOA(526, 390629, F5, 3, 4) (dual of [(390629, 3), 1171861, 5]-NRT-code), using
(26−4, 26, 781258)-Net over F5 — Digital
Digital (22, 26, 781258)-net over F5, using
- net defined by OOA [i] based on linear OOA(526, 781258, F5, 4, 4) (dual of [(781258, 4), 3125006, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(526, 781258, F5, 3, 4) (dual of [(781258, 3), 2343748, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(526, 781258, F5, 4) (dual of [781258, 781232, 5]-code), using
- trace code [i] based on linear OA(2513, 390629, F25, 4) (dual of [390629, 390616, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(2513, 390625, F25, 4) (dual of [390625, 390612, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(259, 390625, F25, 3) (dual of [390625, 390616, 4]-code or 390625-cap in PG(8,25)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(250, 4, F25, 0) (dual of [4, 4, 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(3) ⊂ Ce(2) [i] based on
- trace code [i] based on linear OA(2513, 390629, F25, 4) (dual of [390629, 390616, 5]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(526, 781258, F5, 4) (dual of [781258, 781232, 5]-code), using
- appending kth column [i] based on linear OOA(526, 781258, F5, 3, 4) (dual of [(781258, 3), 2343748, 5]-NRT-code), using
(26−4, 26, large)-Net in Base 5 — Upper bound on s
There is no (22, 26, large)-net in base 5, because
- 2 times m-reduction [i] would yield (22, 24, large)-net in base 5, but