Best Known (28, 28+15, s)-Nets in Base 49
(28, 28+15, 16807)-Net over F49 — Constructive and digital
Digital (28, 43, 16807)-net over F49, using
- net defined by OOA [i] based on linear OOA(4943, 16807, F49, 15, 15) (dual of [(16807, 15), 252062, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4943, 117650, F49, 15) (dual of [117650, 117607, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(4943, 117650, F49, 15) (dual of [117650, 117607, 16]-code), using
(28, 28+15, 58826)-Net over F49 — Digital
Digital (28, 43, 58826)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4943, 58826, F49, 2, 15) (dual of [(58826, 2), 117609, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4943, 117652, F49, 15) (dual of [117652, 117609, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(4943, 117649, F49, 15) (dual of [117649, 117606, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(4940, 117649, F49, 14) (dual of [117649, 117609, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(14) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(4943, 117652, F49, 15) (dual of [117652, 117609, 16]-code), using
(28, 28+15, large)-Net in Base 49 — Upper bound on s
There is no (28, 43, large)-net in base 49, because
- 13 times m-reduction [i] would yield (28, 30, large)-net in base 49, but