Best Known (27−13, 27, s)-Nets in Base 25
(27−13, 27, 132)-Net over F25 — Constructive and digital
Digital (14, 27, 132)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (4, 10, 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, 17, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25 (see above)
- digital (4, 10, 66)-net over F25, using
(27−13, 27, 407)-Net over F25 — Digital
Digital (14, 27, 407)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2527, 407, F25, 13) (dual of [407, 380, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2527, 633, F25, 13) (dual of [633, 606, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(2525, 625, F25, 13) (dual of [625, 600, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2519, 625, F25, 10) (dual of [625, 606, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(2527, 633, F25, 13) (dual of [633, 606, 14]-code), using
(27−13, 27, 142476)-Net in Base 25 — Upper bound on s
There is no (14, 27, 142477)-net in base 25, because
- 1 times m-reduction [i] would yield (14, 26, 142477)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 2 220515 556649 033769 997930 835489 728145 > 2526 [i]