Best Known (10−4, 10, s)-Nets in Base 49
(10−4, 10, 58826)-Net over F49 — Constructive and digital
Digital (6, 10, 58826)-net over F49, using
- net defined by OOA [i] based on linear OOA(4910, 58826, F49, 4, 4) (dual of [(58826, 4), 235294, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(4910, 58826, F49, 3, 4) (dual of [(58826, 3), 176468, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(4910, 117652, F49, 4) (dual of [117652, 117642, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(4910, 117649, F49, 4) (dual of [117649, 117639, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(497, 117649, F49, 3) (dual of [117649, 117642, 4]-code or 117649-cap in PG(6,49)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(490, 3, F49, 0) (dual of [3, 3, 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
- OA 2-folding and stacking [i] based on linear OA(4910, 117652, F49, 4) (dual of [117652, 117642, 5]-code), using
- appending kth column [i] based on linear OOA(4910, 58826, F49, 3, 4) (dual of [(58826, 3), 176468, 5]-NRT-code), using
(10−4, 10, 117652)-Net over F49 — Digital
Digital (6, 10, 117652)-net over F49, using
- net defined by OOA [i] based on linear OOA(4910, 117652, F49, 4, 4) (dual of [(117652, 4), 470598, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(4910, 117652, F49, 3, 4) (dual of [(117652, 3), 352946, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4910, 117652, F49, 4) (dual of [117652, 117642, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(4910, 117649, F49, 4) (dual of [117649, 117639, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(497, 117649, F49, 3) (dual of [117649, 117642, 4]-code or 117649-cap in PG(6,49)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(490, 3, F49, 0) (dual of [3, 3, 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(4910, 117652, F49, 4) (dual of [117652, 117642, 5]-code), using
- appending kth column [i] based on linear OOA(4910, 117652, F49, 3, 4) (dual of [(117652, 3), 352946, 5]-NRT-code), using
(10−4, 10, 8322506)-Net in Base 49 — Upper bound on s
There is no (6, 10, 8322507)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 79792 279811 825185 > 4910 [i]