Best Known (21−10, 21, s)-Nets in Base 49
(21−10, 21, 481)-Net over F49 — Constructive and digital
Digital (11, 21, 481)-net over F49, using
- 1 times m-reduction [i] based on digital (11, 22, 481)-net over F49, using
- net defined by OOA [i] based on linear OOA(4922, 481, F49, 11, 11) (dual of [(481, 11), 5269, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(4922, 2406, F49, 11) (dual of [2406, 2384, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(4922, 2407, F49, 11) (dual of [2407, 2385, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 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(4917, 2402, F49, 9) (dual of [2402, 2385, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4922, 2407, F49, 11) (dual of [2407, 2385, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(4922, 2406, F49, 11) (dual of [2406, 2384, 12]-code), using
- net defined by OOA [i] based on linear OOA(4922, 481, F49, 11, 11) (dual of [(481, 11), 5269, 12]-NRT-code), using
(21−10, 21, 1315)-Net over F49 — Digital
Digital (11, 21, 1315)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4921, 1315, F49, 10) (dual of [1315, 1294, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(4921, 2409, F49, 10) (dual of [2409, 2388, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(4919, 2401, F49, 10) (dual of [2401, 2382, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(4913, 2401, F49, 7) (dual of [2401, 2388, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(9) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(4921, 2409, F49, 10) (dual of [2409, 2388, 11]-code), using
(21−10, 21, 681423)-Net in Base 49 — Upper bound on s
There is no (11, 21, 681424)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 311974 267471 493526 614389 741188 023041 > 4921 [i]