Best Known (37, 37+13, s)-Nets in Base 49
(37, 37+13, 960801)-Net over F49 — Constructive and digital
Digital (37, 50, 960801)-net over F49, using
- net defined by OOA [i] based on linear OOA(4950, 960801, F49, 13, 13) (dual of [(960801, 13), 12490363, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4950, 5764807, F49, 13) (dual of [5764807, 5764757, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4950, 5764811, F49, 13) (dual of [5764811, 5764761, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(4949, 5764802, F49, 13) (dual of [5764802, 5764753, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4941, 5764802, F49, 11) (dual of [5764802, 5764761, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(491, 9, F49, 1) (dual of [9, 8, 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
- discarding factors / shortening the dual code based on linear OA(4950, 5764811, F49, 13) (dual of [5764811, 5764761, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4950, 5764807, F49, 13) (dual of [5764807, 5764757, 14]-code), using
(37, 37+13, 3458021)-Net over F49 — Digital
Digital (37, 50, 3458021)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4950, 3458021, F49, 13) (dual of [3458021, 3457971, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4950, 5764811, F49, 13) (dual of [5764811, 5764761, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(4949, 5764802, F49, 13) (dual of [5764802, 5764753, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4941, 5764802, F49, 11) (dual of [5764802, 5764761, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(491, 9, F49, 1) (dual of [9, 8, 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
- discarding factors / shortening the dual code based on linear OA(4950, 5764811, F49, 13) (dual of [5764811, 5764761, 14]-code), using
(37, 37+13, large)-Net in Base 49 — Upper bound on s
There is no (37, 50, large)-net in base 49, because
- 11 times m-reduction [i] would yield (37, 39, large)-net in base 49, but