Best Known (17, 34, s)-Nets in Base 49
(17, 34, 300)-Net over F49 — Constructive and digital
Digital (17, 34, 300)-net over F49, using
- 491 times duplication [i] based on digital (16, 33, 300)-net over F49, using
- net defined by OOA [i] based on linear OOA(4933, 300, F49, 17, 17) (dual of [(300, 17), 5067, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4933, 2401, F49, 17) (dual of [2401, 2368, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(4933, 2401, F49, 17) (dual of [2401, 2368, 18]-code), using
- net defined by OOA [i] based on linear OOA(4933, 300, F49, 17, 17) (dual of [(300, 17), 5067, 18]-NRT-code), using
(17, 34, 913)-Net over F49 — Digital
Digital (17, 34, 913)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4934, 913, F49, 2, 17) (dual of [(913, 2), 1792, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4934, 1203, F49, 2, 17) (dual of [(1203, 2), 2372, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4934, 2406, F49, 17) (dual of [2406, 2372, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4934, 2407, F49, 17) (dual of [2407, 2373, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(4933, 2402, F49, 17) (dual of [2402, 2369, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(4929, 2402, F49, 15) (dual of [2402, 2373, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,8]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4934, 2407, F49, 17) (dual of [2407, 2373, 18]-code), using
- OOA 2-folding [i] based on linear OA(4934, 2406, F49, 17) (dual of [2406, 2372, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(4934, 1203, F49, 2, 17) (dual of [(1203, 2), 2372, 18]-NRT-code), using
(17, 34, 735369)-Net in Base 49 — Upper bound on s
There is no (17, 34, 735370)-net in base 49, because
- 1 times m-reduction [i] would yield (17, 33, 735370)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 59 768721 665187 365494 515972 266543 046551 400264 519565 308673 > 4933 [i]