Best Known (31, 42, s)-Nets in Base 49
(31, 42, 1152962)-Net over F49 — Constructive and digital
Digital (31, 42, 1152962)-net over F49, using
- net defined by OOA [i] based on linear OOA(4942, 1152962, F49, 11, 11) (dual of [(1152962, 11), 12682540, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(4942, 5764811, F49, 11) (dual of [5764811, 5764769, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 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(4933, 5764802, F49, 9) (dual of [5764802, 5764769, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,5]) ⊂ C([0,4]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(4942, 5764811, F49, 11) (dual of [5764811, 5764769, 12]-code), using
(31, 42, 4328047)-Net over F49 — Digital
Digital (31, 42, 4328047)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4942, 4328047, F49, 11) (dual of [4328047, 4328005, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(4942, 5764811, F49, 11) (dual of [5764811, 5764769, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 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(4933, 5764802, F49, 9) (dual of [5764802, 5764769, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4942, 5764811, F49, 11) (dual of [5764811, 5764769, 12]-code), using
(31, 42, large)-Net in Base 49 — Upper bound on s
There is no (31, 42, large)-net in base 49, because
- 9 times m-reduction [i] would yield (31, 33, large)-net in base 49, but