Best Known (12−6, 12, s)-Nets in Base 49
(12−6, 12, 802)-Net over F49 — Constructive and digital
Digital (6, 12, 802)-net over F49, using
- net defined by OOA [i] based on linear OOA(4912, 802, F49, 6, 6) (dual of [(802, 6), 4800, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(4912, 2406, F49, 6) (dual of [2406, 2394, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(4911, 2401, F49, 6) (dual of [2401, 2390, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(497, 2401, F49, 4) (dual of [2401, 2394, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(491, 5, F49, 1) (dual of [5, 4, 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(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(4912, 2406, F49, 6) (dual of [2406, 2394, 7]-code), using
(12−6, 12, 2049)-Net over F49 — Digital
Digital (6, 12, 2049)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4912, 2049, F49, 6) (dual of [2049, 2037, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(4912, 2406, F49, 6) (dual of [2406, 2394, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(4911, 2401, F49, 6) (dual of [2401, 2390, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(497, 2401, F49, 4) (dual of [2401, 2394, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(491, 5, F49, 1) (dual of [5, 4, 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(5) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(4912, 2406, F49, 6) (dual of [2406, 2394, 7]-code), using
(12−6, 12, 218235)-Net in Base 49 — Upper bound on s
There is no (6, 12, 218236)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 191 583447 399307 159489 > 4912 [i]