Best Known (68, 68+21, s)-Nets in Base 25
(68, 68+21, 39065)-Net over F25 — Constructive and digital
Digital (68, 89, 39065)-net over F25, using
- 253 times duplication [i] based on digital (65, 86, 39065)-net over F25, using
- net defined by OOA [i] based on linear OOA(2586, 39065, F25, 21, 21) (dual of [(39065, 21), 820279, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2586, 390651, F25, 21) (dual of [390651, 390565, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2586, 390652, F25, 21) (dual of [390652, 390566, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(2581, 390626, F25, 21) (dual of [390626, 390545, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 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(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([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2586, 390652, F25, 21) (dual of [390652, 390566, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2586, 390651, F25, 21) (dual of [390651, 390565, 22]-code), using
- net defined by OOA [i] based on linear OOA(2586, 39065, F25, 21, 21) (dual of [(39065, 21), 820279, 22]-NRT-code), using
(68, 68+21, 575355)-Net over F25 — Digital
Digital (68, 89, 575355)-net over F25, using
(68, 68+21, large)-Net in Base 25 — Upper bound on s
There is no (68, 89, large)-net in base 25, because
- 19 times m-reduction [i] would yield (68, 70, large)-net in base 25, but