Best Known (58−15, 58, s)-Nets in Base 25
(58−15, 58, 55804)-Net over F25 — Constructive and digital
Digital (43, 58, 55804)-net over F25, using
- 251 times duplication [i] based on digital (42, 57, 55804)-net over F25, using
- net defined by OOA [i] based on linear OOA(2557, 55804, F25, 15, 15) (dual of [(55804, 15), 837003, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2557, 390629, F25, 15) (dual of [390629, 390572, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(2557, 390625, F25, 15) (dual of [390625, 390568, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2553, 390625, F25, 14) (dual of [390625, 390572, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(14) ⊂ Ce(13) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(2557, 390629, F25, 15) (dual of [390629, 390572, 16]-code), using
- net defined by OOA [i] based on linear OOA(2557, 55804, F25, 15, 15) (dual of [(55804, 15), 837003, 16]-NRT-code), using
(58−15, 58, 318138)-Net over F25 — Digital
Digital (43, 58, 318138)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2558, 318138, F25, 15) (dual of [318138, 318080, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2558, 390635, F25, 15) (dual of [390635, 390577, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(2557, 390626, F25, 15) (dual of [390626, 390569, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(2549, 390626, F25, 13) (dual of [390626, 390577, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(251, 9, F25, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2558, 390635, F25, 15) (dual of [390635, 390577, 16]-code), using
(58−15, 58, large)-Net in Base 25 — Upper bound on s
There is no (43, 58, large)-net in base 25, because
- 13 times m-reduction [i] would yield (43, 45, large)-net in base 25, but