Best Known (29, 44, s)-Nets in Base 49
(29, 44, 16808)-Net over F49 — Constructive and digital
Digital (29, 44, 16808)-net over F49, using
- net defined by OOA [i] based on linear OOA(4944, 16808, F49, 15, 15) (dual of [(16808, 15), 252076, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4944, 117657, F49, 15) (dual of [117657, 117613, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(4943, 117650, F49, 15) (dual of [117650, 117607, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(4937, 117650, F49, 13) (dual of [117650, 117613, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(491, 7, F49, 1) (dual of [7, 6, 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([0,7]) ⊂ C([0,6]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(4944, 117657, F49, 15) (dual of [117657, 117613, 16]-code), using
(29, 44, 58828)-Net over F49 — Digital
Digital (29, 44, 58828)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4944, 58828, F49, 2, 15) (dual of [(58828, 2), 117612, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4944, 117656, F49, 15) (dual of [117656, 117612, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4944, 117657, F49, 15) (dual of [117657, 117613, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(4943, 117650, F49, 15) (dual of [117650, 117607, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(4937, 117650, F49, 13) (dual of [117650, 117613, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(491, 7, F49, 1) (dual of [7, 6, 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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4944, 117657, F49, 15) (dual of [117657, 117613, 16]-code), using
- OOA 2-folding [i] based on linear OA(4944, 117656, F49, 15) (dual of [117656, 117612, 16]-code), using
(29, 44, large)-Net in Base 49 — Upper bound on s
There is no (29, 44, large)-net in base 49, because
- 13 times m-reduction [i] would yield (29, 31, large)-net in base 49, but