Best Known (20, 20+5, s)-Nets in Base 25
(20, 20+5, 4882850)-Net over F25 — Constructive and digital
Digital (20, 25, 4882850)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 195314)-net over F25, using
- s-reduction based on digital (0, 0, s)-net over F25 with arbitrarily large s, using
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 0, 195314)-net over F25 (see above)
- digital (0, 1, 195314)-net over F25, using
- s-reduction based on digital (0, 1, s)-net over F25 with arbitrarily large s, using
- digital (0, 1, 195314)-net over F25 (see above)
- digital (0, 1, 195314)-net over F25 (see above)
- digital (3, 5, 195314)-net over F25, using
- s-reduction based on digital (3, 5, 406901)-net over F25, using
- digital (12, 17, 195314)-net over F25, using
- net defined by OOA [i] based on linear OOA(2517, 195314, F25, 5, 5) (dual of [(195314, 5), 976553, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2517, 390629, F25, 5) (dual of [390629, 390612, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(2517, 390625, F25, 5) (dual of [390625, 390608, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- 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(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(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(2517, 390629, F25, 5) (dual of [390629, 390612, 6]-code), using
- net defined by OOA [i] based on linear OOA(2517, 195314, F25, 5, 5) (dual of [(195314, 5), 976553, 6]-NRT-code), using
- digital (0, 0, 195314)-net over F25, using
(20, 20+5, large)-Net over F25 — Digital
Digital (20, 25, large)-net over F25, using
- 1 times m-reduction [i] based on digital (20, 26, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2526, large, F25, 6) (dual of [large, large−26, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2526, large, F25, 6) (dual of [large, large−26, 7]-code), using
(20, 20+5, large)-Net in Base 25 — Upper bound on s
There is no (20, 25, large)-net in base 25, because
- 3 times m-reduction [i] would yield (20, 22, large)-net in base 25, but