Best Known (8, 13, s)-Nets in Base 49
(8, 13, 58825)-Net over F49 — Constructive and digital
Digital (8, 13, 58825)-net over F49, using
- net defined by OOA [i] based on linear OOA(4913, 58825, F49, 5, 5) (dual of [(58825, 5), 294112, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(4913, 117651, F49, 5) (dual of [117651, 117638, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(4913, 117652, F49, 5) (dual of [117652, 117639, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(4913, 117649, F49, 5) (dual of [117649, 117636, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- 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(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(4) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(4913, 117652, F49, 5) (dual of [117652, 117639, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(4913, 117651, F49, 5) (dual of [117651, 117638, 6]-code), using
(8, 13, 117652)-Net over F49 — Digital
Digital (8, 13, 117652)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4913, 117652, F49, 5) (dual of [117652, 117639, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(4913, 117649, F49, 5) (dual of [117649, 117636, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- 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(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(4) ⊂ Ce(3) [i] based on
(8, 13, large)-Net in Base 49 — Upper bound on s
There is no (8, 13, large)-net in base 49, because
- 3 times m-reduction [i] would yield (8, 10, large)-net in base 49, but