Best Known (24, 24+13, s)-Nets in Base 49
(24, 24+13, 19608)-Net over F49 — Constructive and digital
Digital (24, 37, 19608)-net over F49, using
- net defined by OOA [i] based on linear OOA(4937, 19608, F49, 13, 13) (dual of [(19608, 13), 254867, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4937, 117649, F49, 13) (dual of [117649, 117612, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 6-folding and stacking with additional row [i] based on linear OA(4937, 117649, F49, 13) (dual of [117649, 117612, 14]-code), using
(24, 24+13, 58826)-Net over F49 — Digital
Digital (24, 37, 58826)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4937, 58826, F49, 2, 13) (dual of [(58826, 2), 117615, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4937, 117652, F49, 13) (dual of [117652, 117615, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(4937, 117649, F49, 13) (dual of [117649, 117612, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(4934, 117649, F49, 12) (dual of [117649, 117615, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- OOA 2-folding [i] based on linear OA(4937, 117652, F49, 13) (dual of [117652, 117615, 14]-code), using
(24, 24+13, large)-Net in Base 49 — Upper bound on s
There is no (24, 37, large)-net in base 49, because
- 11 times m-reduction [i] would yield (24, 26, large)-net in base 49, but