Best Known (33, 64, s)-Nets in Base 25
(33, 64, 208)-Net over F25 — Constructive and digital
Digital (33, 64, 208)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (9, 24, 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, 40, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25 (see above)
- digital (9, 24, 104)-net over F25, using
(33, 64, 516)-Net over F25 — Digital
Digital (33, 64, 516)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2564, 516, F25, 31) (dual of [516, 452, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2564, 637, F25, 31) (dual of [637, 573, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(2561, 626, F25, 31) (dual of [626, 565, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- 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(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([0,15]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2564, 637, F25, 31) (dual of [637, 573, 32]-code), using
(33, 64, 199015)-Net in Base 25 — Upper bound on s
There is no (33, 64, 199016)-net in base 25, because
- 1 times m-reduction [i] would yield (33, 63, 199016)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 11755 429329 129334 859855 031661 635968 340399 690008 032486 871022 541900 268417 718554 616937 328449 > 2563 [i]