Best Known (3, 3+4, s)-Nets in Base 49
(3, 3+4, 1201)-Net over F49 — Constructive and digital
Digital (3, 7, 1201)-net over F49, using
- net defined by OOA [i] based on linear OOA(497, 1201, F49, 4, 4) (dual of [(1201, 4), 4797, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(497, 1201, F49, 3, 4) (dual of [(1201, 3), 3596, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(497, 2402, F49, 4) (dual of [2402, 2395, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(497, 2403, F49, 4) (dual of [2403, 2396, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- 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(495, 2401, F49, 3) (dual of [2401, 2396, 4]-code or 2401-cap in PG(4,49)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(490, 2, F49, 0) (dual of [2, 2, 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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(497, 2403, F49, 4) (dual of [2403, 2396, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(497, 2402, F49, 4) (dual of [2402, 2395, 5]-code), using
- appending kth column [i] based on linear OOA(497, 1201, F49, 3, 4) (dual of [(1201, 3), 3596, 5]-NRT-code), using
(3, 3+4, 2403)-Net over F49 — Digital
Digital (3, 7, 2403)-net over F49, using
- net defined by OOA [i] based on linear OOA(497, 2403, F49, 4, 4) (dual of [(2403, 4), 9605, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(497, 2403, F49, 3, 4) (dual of [(2403, 3), 7202, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(497, 2403, F49, 4) (dual of [2403, 2396, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- 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(495, 2401, F49, 3) (dual of [2401, 2396, 4]-code or 2401-cap in PG(4,49)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(490, 2, F49, 0) (dual of [2, 2, 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(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(497, 2403, F49, 4) (dual of [2403, 2396, 5]-code), using
- appending kth column [i] based on linear OOA(497, 2403, F49, 3, 4) (dual of [(2403, 3), 7202, 5]-NRT-code), using
(3, 3+4, 24263)-Net in Base 49 — Upper bound on s
There is no (3, 7, 24264)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 678260 715265 > 497 [i]