Best Known (31, 31+27, s)-Nets in Base 25
(31, 31+27, 208)-Net over F25 — Constructive and digital
Digital (31, 58, 208)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (9, 22, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- digital (9, 36, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25 (see above)
- digital (9, 22, 104)-net over F25, using
(31, 31+27, 641)-Net over F25 — Digital
Digital (31, 58, 641)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2558, 641, F25, 27) (dual of [641, 583, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2558, 643, F25, 27) (dual of [643, 585, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(2553, 626, F25, 27) (dual of [626, 573, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- 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(255, 17, F25, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,25)), using
- discarding factors / shortening the dual code based on linear OA(255, 25, F25, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
- Reed–Solomon code RS(20,25) [i]
- discarding factors / shortening the dual code based on linear OA(255, 25, F25, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2558, 643, F25, 27) (dual of [643, 585, 28]-code), using
(31, 31+27, 318137)-Net in Base 25 — Upper bound on s
There is no (31, 58, 318138)-net in base 25, because
- 1 times m-reduction [i] would yield (31, 57, 318138)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 48 148639 529271 981654 051776 779590 929766 422563 957346 868600 112080 809965 515730 987825 > 2557 [i]