Best Known (32−9, 32, s)-Nets in Base 49
(32−9, 32, 30613)-Net over F49 — Constructive and digital
Digital (23, 32, 30613)-net over F49, using
- (u, u+v)-construction [i] based on
- 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
- net defined by OOA [i] based on linear OOA(497, 1201, F49, 4, 4) (dual of [(1201, 4), 4797, 5]-NRT-code), using
- digital (16, 25, 29412)-net over F49, using
- net defined by OOA [i] based on linear OOA(4925, 29412, F49, 9, 9) (dual of [(29412, 9), 264683, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(4925, 117649, F49, 9) (dual of [117649, 117624, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(4925, 117649, F49, 9) (dual of [117649, 117624, 10]-code), using
- net defined by OOA [i] based on linear OOA(4925, 29412, F49, 9, 9) (dual of [(29412, 9), 264683, 10]-NRT-code), using
- digital (3, 7, 1201)-net over F49, using
(32−9, 32, 452103)-Net over F49 — Digital
Digital (23, 32, 452103)-net over F49, using
(32−9, 32, large)-Net in Base 49 — Upper bound on s
There is no (23, 32, large)-net in base 49, because
- 7 times m-reduction [i] would yield (23, 25, large)-net in base 49, but