Best Known (55, 55+8, s)-Nets in Base 25
(55, 55+8, 6784686)-Net over F25 — Constructive and digital
Digital (55, 63, 6784686)-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, 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 (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 (3, 5, 195314)-net over F25 (see above)
- digital (9, 13, 195314)-net over F25, using
- net defined by OOA [i] based on linear OOA(2513, 195314, F25, 4, 4) (dual of [(195314, 4), 781243, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2513, 390628, F25, 4) (dual of [390628, 390615, 5]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code 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(2513, 390628, F25, 4) (dual of [390628, 390615, 5]-code), using
- net defined by OOA [i] based on linear OOA(2513, 195314, F25, 4, 4) (dual of [(195314, 4), 781243, 5]-NRT-code), using
- digital (28, 36, 2097150)-net over F25, using
- net defined by OOA [i] based on linear OOA(2536, 2097150, F25, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2536, 8388600, F25, 8) (dual of [8388600, 8388564, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(2536, 8388600, F25, 8) (dual of [8388600, 8388564, 9]-code), using
- net defined by OOA [i] based on linear OOA(2536, 2097150, F25, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
- digital (0, 0, 195314)-net over F25, using
(55, 55+8, large)-Net over F25 — Digital
Digital (55, 63, large)-net over F25, using
- t-expansion [i] based on digital (53, 63, large)-net over F25, using
- 4 times m-reduction [i] based on digital (53, 67, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2567, large, F25, 14) (dual of [large, large−67, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2566, large, F25, 14) (dual of [large, large−66, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 1 times code embedding in larger space [i] based on linear OA(2566, large, F25, 14) (dual of [large, large−66, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2567, large, F25, 14) (dual of [large, large−67, 15]-code), using
- 4 times m-reduction [i] based on digital (53, 67, large)-net over F25, using
(55, 55+8, large)-Net in Base 25 — Upper bound on s
There is no (55, 63, large)-net in base 25, because
- 6 times m-reduction [i] would yield (55, 57, large)-net in base 25, but