Best Known (15, 15+12, s)-Nets in Base 25
(15, 15+12, 132)-Net over F25 — Constructive and digital
Digital (15, 27, 132)-net over F25, using
- 2 times m-reduction [i] based on digital (15, 29, 132)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 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, 18, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25 (see above)
- digital (4, 11, 66)-net over F25, using
- (u, u+v)-construction [i] based on
(15, 15+12, 669)-Net over F25 — Digital
Digital (15, 27, 669)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2527, 669, F25, 12) (dual of [669, 642, 13]-code), using
- 35 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 28 times 0) [i] based on linear OA(2524, 631, F25, 12) (dual of [631, 607, 13]-code), using
- construction XX applied to C1 = C([622,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([622,9]) [i] based on
- linear OA(2521, 624, F25, 11) (dual of [624, 603, 12]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2519, 624, F25, 10) (dual of [624, 605, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2523, 624, F25, 12) (dual of [624, 601, 13]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−2,−1,…,9}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2517, 624, F25, 9) (dual of [624, 607, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [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, 25, F25, 1) (dual of [25, 24, 2]-code), using
- Reed–Solomon code RS(24,25) [i]
- discarding factors / shortening the dual code based on linear OA(251, 25, F25, 1) (dual of [25, 24, 2]-code), using
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([622,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([622,9]) [i] based on
- 35 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 28 times 0) [i] based on linear OA(2524, 631, F25, 12) (dual of [631, 607, 13]-code), using
(15, 15+12, 243632)-Net in Base 25 — Upper bound on s
There is no (15, 27, 243633)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 55 511254 737203 721846 576096 424355 474257 > 2527 [i]