Best Known (45−12, 45, s)-Nets in Base 49
(45−12, 45, 960800)-Net over F49 — Constructive and digital
Digital (33, 45, 960800)-net over F49, using
- net defined by OOA [i] based on linear OOA(4945, 960800, F49, 12, 12) (dual of [(960800, 12), 11529555, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(4945, 5764800, F49, 12) (dual of [5764800, 5764755, 13]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(4945, 5764801, F49, 12) (dual of [5764801, 5764756, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(4945, 5764800, F49, 12) (dual of [5764800, 5764755, 13]-code), using
(45−12, 45, 2882402)-Net over F49 — Digital
Digital (33, 45, 2882402)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4945, 2882402, F49, 2, 12) (dual of [(2882402, 2), 5764759, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4945, 5764804, F49, 12) (dual of [5764804, 5764759, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(4945, 5764805, F49, 12) (dual of [5764805, 5764760, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 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(4941, 5764801, F49, 11) (dual of [5764801, 5764760, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(490, 4, F49, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(4945, 5764805, F49, 12) (dual of [5764805, 5764760, 13]-code), using
- OOA 2-folding [i] based on linear OA(4945, 5764804, F49, 12) (dual of [5764804, 5764759, 13]-code), using
(45−12, 45, large)-Net in Base 49 — Upper bound on s
There is no (33, 45, large)-net in base 49, because
- 10 times m-reduction [i] would yield (33, 35, large)-net in base 49, but