Best Known (45, 45+9, s)-Nets in Base 25
(45, 45+9, 2292464)-Net over F25 — Constructive and digital
Digital (45, 54, 2292464)-net over F25, using
- (u, u+v)-construction [i] based on
- 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 (32, 41, 2097150)-net over F25, using
- net defined by OOA [i] based on linear OOA(2541, 2097150, F25, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2541, 8388601, F25, 9) (dual of [8388601, 8388560, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2541, large, F25, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2541, large, F25, 9) (dual of [large, large−41, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2541, 8388601, F25, 9) (dual of [8388601, 8388560, 10]-code), using
- net defined by OOA [i] based on linear OOA(2541, 2097150, F25, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- digital (9, 13, 195314)-net over F25, using
(45, 45+9, large)-Net over F25 — Digital
Digital (45, 54, large)-net over F25, using
- t-expansion [i] based on digital (44, 54, large)-net over F25, using
- 2 times m-reduction [i] based on digital (44, 56, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2556, large, F25, 12) (dual of [large, large−56, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2556, large, F25, 12) (dual of [large, large−56, 13]-code), using
- 2 times m-reduction [i] based on digital (44, 56, large)-net over F25, using
(45, 45+9, large)-Net in Base 25 — Upper bound on s
There is no (45, 54, large)-net in base 25, because
- 7 times m-reduction [i] would yield (45, 47, large)-net in base 25, but