Best Known (9, 18, s)-Nets in Base 25
(9, 18, 157)-Net over F25 — Constructive and digital
Digital (9, 18, 157)-net over F25, using
- net defined by OOA [i] based on linear OOA(2518, 157, F25, 9, 9) (dual of [(157, 9), 1395, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2518, 629, F25, 9) (dual of [629, 611, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2518, 631, F25, 9) (dual of [631, 613, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(2517, 626, F25, 9) (dual of [626, 609, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(2513, 626, F25, 7) (dual of [626, 613, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [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, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2518, 631, F25, 9) (dual of [631, 613, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2518, 629, F25, 9) (dual of [629, 611, 10]-code), using
(9, 18, 347)-Net over F25 — Digital
Digital (9, 18, 347)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2518, 347, F25, 9) (dual of [347, 329, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2518, 624, F25, 9) (dual of [624, 606, 10]-code), using
(9, 18, 80552)-Net in Base 25 — Upper bound on s
There is no (9, 18, 80553)-net in base 25, because
- 1 times m-reduction [i] would yield (9, 17, 80553)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 582098 586181 643733 184225 > 2517 [i]