Best Known (9, 9+12, s)-Nets in Base 25
(9, 9+12, 104)-Net over F25 — Constructive and digital
Digital (9, 21, 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
(9, 9+12, 108)-Net over F25 — Digital
Digital (9, 21, 108)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2521, 108, F25, 12) (dual of [108, 87, 13]-code), using
- construction XX applied to C1 = C([8,18]), C2 = C([7,17]), C3 = C1 + C2 = C([8,17]), and C∩ = C1 ∩ C2 = C([7,18]) [i] based on
- linear OA(2519, 104, F25, 11) (dual of [104, 85, 12]-code), using the BCH-code C(I) with length 104 | 252−1, defining interval I = {8,9,…,18}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2519, 104, F25, 11) (dual of [104, 85, 12]-code), using the BCH-code C(I) with length 104 | 252−1, defining interval I = {7,8,…,17}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2521, 104, F25, 12) (dual of [104, 83, 13]-code), using the BCH-code C(I) with length 104 | 252−1, defining interval I = {7,8,…,18}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2517, 104, F25, 10) (dual of [104, 87, 11]-code), using the BCH-code C(I) with length 104 | 252−1, defining interval I = {8,9,…,17}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([8,18]), C2 = C([7,17]), C3 = C1 + C2 = C([8,17]), and C∩ = C1 ∩ C2 = C([7,18]) [i] based on
(9, 9+12, 9742)-Net in Base 25 — Upper bound on s
There is no (9, 21, 9743)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 227418 869030 908362 897729 878641 > 2521 [i]