Best Known (38−13, 38, s)-Nets in Base 49
(38−13, 38, 19609)-Net over F49 — Constructive and digital
Digital (25, 38, 19609)-net over F49, using
- net defined by OOA [i] based on linear OOA(4938, 19609, F49, 13, 13) (dual of [(19609, 13), 254879, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4938, 117655, F49, 13) (dual of [117655, 117617, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- 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(4931, 117650, F49, 11) (dual of [117650, 117619, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4938, 117655, F49, 13) (dual of [117655, 117617, 14]-code), using
(38−13, 38, 58828)-Net over F49 — Digital
Digital (25, 38, 58828)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4938, 58828, F49, 2, 13) (dual of [(58828, 2), 117618, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4938, 117656, F49, 13) (dual of [117656, 117618, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- 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(4931, 117650, F49, 11) (dual of [117650, 117619, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4938, 117657, F49, 13) (dual of [117657, 117619, 14]-code), using
- OOA 2-folding [i] based on linear OA(4938, 117656, F49, 13) (dual of [117656, 117618, 14]-code), using
(38−13, 38, large)-Net in Base 49 — Upper bound on s
There is no (25, 38, large)-net in base 49, because
- 11 times m-reduction [i] would yield (25, 27, large)-net in base 49, but