Best Known (25−12, 25, s)-Nets in Base 49
(25−12, 25, 401)-Net over F49 — Constructive and digital
Digital (13, 25, 401)-net over F49, using
- 1 times m-reduction [i] based on digital (13, 26, 401)-net over F49, using
- net defined by OOA [i] based on linear OOA(4926, 401, F49, 13, 13) (dual of [(401, 13), 5187, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4926, 2407, F49, 13) (dual of [2407, 2381, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(4925, 2402, F49, 13) (dual of [2402, 2377, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4921, 2402, F49, 11) (dual of [2402, 2381, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(491, 5, F49, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(4926, 2407, F49, 13) (dual of [2407, 2381, 14]-code), using
- net defined by OOA [i] based on linear OOA(4926, 401, F49, 13, 13) (dual of [(401, 13), 5187, 14]-NRT-code), using
(25−12, 25, 1204)-Net over F49 — Digital
Digital (13, 25, 1204)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4925, 1204, F49, 2, 12) (dual of [(1204, 2), 2383, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4925, 2408, F49, 12) (dual of [2408, 2383, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(4925, 2409, F49, 12) (dual of [2409, 2384, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(4923, 2401, F49, 12) (dual of [2401, 2378, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(4917, 2401, F49, 9) (dual of [2401, 2384, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(492, 8, F49, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,49)), using
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- Reed–Solomon code RS(47,49) [i]
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(4925, 2409, F49, 12) (dual of [2409, 2384, 13]-code), using
- OOA 2-folding [i] based on linear OA(4925, 2408, F49, 12) (dual of [2408, 2383, 13]-code), using
(25−12, 25, 687801)-Net in Base 49 — Upper bound on s
There is no (13, 25, 687802)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 1 798479 445579 342415 025691 534898 370789 022273 > 4925 [i]