Best Known (57, 57+21, s)-Nets in Base 25
(57, 57+21, 1630)-Net over F25 — Constructive and digital
Digital (57, 78, 1630)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- digital (43, 64, 1564)-net over F25, using
- net defined by OOA [i] based on linear OOA(2564, 1564, F25, 21, 21) (dual of [(1564, 21), 32780, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2564, 15641, F25, 21) (dual of [15641, 15577, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2561, 15626, F25, 21) (dual of [15626, 15565, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2549, 15626, F25, 17) (dual of [15626, 15577, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(253, 15, F25, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,25) or 15-cap in PG(2,25)), using
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- Reed–Solomon code RS(22,25) [i]
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(2564, 15641, F25, 21) (dual of [15641, 15577, 22]-code), using
- net defined by OOA [i] based on linear OOA(2564, 1564, F25, 21, 21) (dual of [(1564, 21), 32780, 22]-NRT-code), using
- digital (4, 14, 66)-net over F25, using
(57, 57+21, 97973)-Net over F25 — Digital
Digital (57, 78, 97973)-net over F25, using
(57, 57+21, large)-Net in Base 25 — Upper bound on s
There is no (57, 78, large)-net in base 25, because
- 19 times m-reduction [i] would yield (57, 59, large)-net in base 25, but