Best Known (44−21, 44, s)-Nets in Base 25
(44−21, 44, 178)-Net over F25 — Constructive and digital
Digital (23, 44, 178)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (3, 13, 52)-net over F25, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- digital (10, 31, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- digital (3, 13, 52)-net over F25, using
(44−21, 44, 473)-Net over F25 — Digital
Digital (23, 44, 473)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2544, 473, F25, 21) (dual of [473, 429, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2544, 637, F25, 21) (dual of [637, 593, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2541, 626, F25, 21) (dual of [626, 585, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2533, 626, F25, 17) (dual of [626, 593, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(253, 11, F25, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,25) or 11-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
- discarding factors / shortening the dual code based on linear OA(2544, 637, F25, 21) (dual of [637, 593, 22]-code), using
(44−21, 44, 193595)-Net in Base 25 — Upper bound on s
There is no (23, 44, 193596)-net in base 25, because
- 1 times m-reduction [i] would yield (23, 43, 193596)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 1 292478 943224 254372 163171 567037 935745 677964 822185 048512 733121 > 2543 [i]