Best Known (34−17, 34, s)-Nets in Base 25
(34−17, 34, 132)-Net over F25 — Constructive and digital
Digital (17, 34, 132)-net over F25, using
- 1 times m-reduction [i] based on digital (17, 35, 132)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 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 (4, 22, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25 (see above)
- digital (4, 13, 66)-net over F25, using
- (u, u+v)-construction [i] based on
(34−17, 34, 315)-Net over F25 — Digital
Digital (17, 34, 315)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2534, 315, F25, 2, 17) (dual of [(315, 2), 596, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2534, 630, F25, 17) (dual of [630, 596, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2534, 631, F25, 17) (dual of [631, 597, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- 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(2529, 626, F25, 15) (dual of [626, 597, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(251, 5, F25, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2534, 631, F25, 17) (dual of [631, 597, 18]-code), using
- OOA 2-folding [i] based on linear OA(2534, 630, F25, 17) (dual of [630, 596, 18]-code), using
(34−17, 34, 91614)-Net in Base 25 — Upper bound on s
There is no (17, 34, 91615)-net in base 25, because
- 1 times m-reduction [i] would yield (17, 33, 91615)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 13553 375361 587340 119159 938804 290223 237821 978433 > 2533 [i]