Best Known (39, 39+37, s)-Nets in Base 25
(39, 39+37, 252)-Net over F25 — Constructive and digital
Digital (39, 76, 252)-net over F25, using
- 1 times m-reduction [i] based on digital (39, 77, 252)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (10, 29, 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 (10, 48, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- digital (10, 29, 126)-net over F25, using
- (u, u+v)-construction [i] based on
(39, 39+37, 558)-Net over F25 — Digital
Digital (39, 76, 558)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2576, 558, F25, 37) (dual of [558, 482, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2576, 637, F25, 37) (dual of [637, 561, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(2573, 626, F25, 37) (dual of [626, 553, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(2565, 626, F25, 33) (dual of [626, 561, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (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([0,18]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2576, 637, F25, 37) (dual of [637, 561, 38]-code), using
(39, 39+37, 210208)-Net in Base 25 — Upper bound on s
There is no (39, 76, 210209)-net in base 25, because
- 1 times m-reduction [i] would yield (39, 75, 210209)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 700 694683 403696 012106 104184 420656 994897 875127 377821 096122 941745 060831 790977 512556 644590 130672 239409 843185 > 2575 [i]