Best Known (31−15, 31, s)-Nets in Base 25
(31−15, 31, 132)-Net over F25 — Constructive and digital
Digital (16, 31, 132)-net over F25, using
- 1 times m-reduction [i] based on digital (16, 32, 132)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 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, 20, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25 (see above)
- digital (4, 12, 66)-net over F25, using
- (u, u+v)-construction [i] based on
(31−15, 31, 392)-Net over F25 — Digital
Digital (16, 31, 392)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2531, 392, F25, 15) (dual of [392, 361, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2531, 633, F25, 15) (dual of [633, 602, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(2529, 625, F25, 15) (dual of [625, 596, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2523, 625, F25, 12) (dual of [625, 602, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(252, 8, F25, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(2531, 633, F25, 15) (dual of [633, 602, 16]-code), using
(31−15, 31, 137996)-Net in Base 25 — Upper bound on s
There is no (16, 31, 137997)-net in base 25, because
- 1 times m-reduction [i] would yield (16, 30, 137997)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 867377 565662 877745 437461 471764 782567 823945 > 2530 [i]