Best Known (59, 59+19, s)-Nets in Base 25
(59, 59+19, 43405)-Net over F25 — Constructive and digital
Digital (59, 78, 43405)-net over F25, using
- 251 times duplication [i] based on digital (58, 77, 43405)-net over F25, using
- net defined by OOA [i] based on linear OOA(2577, 43405, F25, 19, 19) (dual of [(43405, 19), 824618, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2577, 390646, F25, 19) (dual of [390646, 390569, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2577, 390649, F25, 19) (dual of [390649, 390572, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(2573, 390625, F25, 19) (dual of [390625, 390552, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(254, 24, F25, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,25)), using
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- Reed–Solomon code RS(21,25) [i]
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(2577, 390649, F25, 19) (dual of [390649, 390572, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2577, 390646, F25, 19) (dual of [390646, 390569, 20]-code), using
- net defined by OOA [i] based on linear OOA(2577, 43405, F25, 19, 19) (dual of [(43405, 19), 824618, 20]-NRT-code), using
(59, 59+19, 390652)-Net over F25 — Digital
Digital (59, 78, 390652)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2578, 390652, F25, 19) (dual of [390652, 390574, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(2573, 390626, F25, 19) (dual of [390626, 390553, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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(255, 26, F25, 5) (dual of [26, 21, 6]-code or 26-arc in PG(4,25)), using
- extended Reed–Solomon code RSe(21,25) [i]
- the expurgated narrow-sense BCH-code C(I) with length 26 | 252−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- algebraic-geometric code AG(F,10P) with degPÂ =Â 2 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using the rational function field F25(x) [i]
- algebraic-geometric code AG(F, Q+6P) with degQ = 2 and degPÂ =Â 3 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26 (see above)
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
(59, 59+19, large)-Net in Base 25 — Upper bound on s
There is no (59, 78, large)-net in base 25, because
- 17 times m-reduction [i] would yield (59, 61, large)-net in base 25, but