Best Known (27−13, 27, s)-Nets in Base 49
(27−13, 27, 401)-Net over F49 — Constructive and digital
Digital (14, 27, 401)-net over F49, using
- 491 times duplication [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
(27−13, 27, 1204)-Net over F49 — Digital
Digital (14, 27, 1204)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4927, 1204, F49, 2, 13) (dual of [(1204, 2), 2381, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4927, 2408, F49, 13) (dual of [2408, 2381, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4927, 2409, F49, 13) (dual of [2409, 2382, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(4925, 2401, F49, 13) (dual of [2401, 2376, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(4927, 2409, F49, 13) (dual of [2409, 2382, 14]-code), using
- OOA 2-folding [i] based on linear OA(4927, 2408, F49, 13) (dual of [2408, 2381, 14]-code), using
(27−13, 27, 1315718)-Net in Base 49 — Upper bound on s
There is no (14, 27, 1315719)-net in base 49, because
- 1 times m-reduction [i] would yield (14, 26, 1315719)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 88 124974 493792 598313 278032 410898 695665 843169 > 4926 [i]