Best Known (54−14, 54, s)-Nets in Base 49
(54−14, 54, 823544)-Net over F49 — Constructive and digital
Digital (40, 54, 823544)-net over F49, using
- net defined by OOA [i] based on linear OOA(4954, 823544, F49, 14, 14) (dual of [(823544, 14), 11529562, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(4954, 5764808, F49, 14) (dual of [5764808, 5764754, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4954, 5764810, F49, 14) (dual of [5764810, 5764756, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(4953, 5764801, F49, 14) (dual of [5764801, 5764748, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(4945, 5764801, F49, 12) (dual of [5764801, 5764756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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 Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(4954, 5764810, F49, 14) (dual of [5764810, 5764756, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(4954, 5764808, F49, 14) (dual of [5764808, 5764754, 15]-code), using
(54−14, 54, 3214787)-Net over F49 — Digital
Digital (40, 54, 3214787)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4954, 3214787, F49, 14) (dual of [3214787, 3214733, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4954, 5764810, F49, 14) (dual of [5764810, 5764756, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(4953, 5764801, F49, 14) (dual of [5764801, 5764748, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(4945, 5764801, F49, 12) (dual of [5764801, 5764756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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 Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(4954, 5764810, F49, 14) (dual of [5764810, 5764756, 15]-code), using
(54−14, 54, large)-Net in Base 49 — Upper bound on s
There is no (40, 54, large)-net in base 49, because
- 12 times m-reduction [i] would yield (40, 42, large)-net in base 49, but