Best Known (36−15, 36, s)-Nets in Base 49
(36−15, 36, 393)-Net over F49 — Constructive and digital
Digital (21, 36, 393)-net over F49, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- digital (14, 29, 343)-net over F49, using
- net defined by OOA [i] based on linear OOA(4929, 343, F49, 15, 15) (dual of [(343, 15), 5116, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4929, 2402, F49, 15) (dual of [2402, 2373, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, 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(4929, 2402, F49, 15) (dual of [2402, 2373, 16]-code), using
- net defined by OOA [i] based on linear OOA(4929, 343, F49, 15, 15) (dual of [(343, 15), 5116, 16]-NRT-code), using
- digital (0, 7, 50)-net over F49, using
(36−15, 36, 2947)-Net over F49 — Digital
Digital (21, 36, 2947)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4936, 2947, F49, 15) (dual of [2947, 2911, 16]-code), using
- 537 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 8 times 0, 1, 35 times 0, 1, 123 times 0, 1, 365 times 0) [i] based on linear OA(4929, 2403, F49, 15) (dual of [2403, 2374, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(4929, 2401, F49, 15) (dual of [2401, 2372, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(4927, 2401, F49, 14) (dual of [2401, 2374, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(14) ⊂ Ce(13) [i] based on
- 537 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 8 times 0, 1, 35 times 0, 1, 123 times 0, 1, 365 times 0) [i] based on linear OA(4929, 2403, F49, 15) (dual of [2403, 2374, 16]-code), using
(36−15, 36, large)-Net in Base 49 — Upper bound on s
There is no (21, 36, large)-net in base 49, because
- 13 times m-reduction [i] would yield (21, 23, large)-net in base 49, but