Best Known (217−19, 217, s)-Nets in Base 4
(217−19, 217, 1864142)-Net over F4 — Constructive and digital
Digital (198, 217, 1864142)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 11, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (187, 206, 1864132)-net over F4, using
- trace code for nets [i] based on digital (84, 103, 932066)-net over F16, using
- net defined by OOA [i] based on linear OOA(16103, 932066, F16, 19, 19) (dual of [(932066, 19), 17709151, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(16103, 8388595, F16, 19) (dual of [8388595, 8388492, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(16103, large, F16, 19) (dual of [large, large−103, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(16103, large, F16, 19) (dual of [large, large−103, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(16103, 8388595, F16, 19) (dual of [8388595, 8388492, 20]-code), using
- net defined by OOA [i] based on linear OOA(16103, 932066, F16, 19, 19) (dual of [(932066, 19), 17709151, 20]-NRT-code), using
- trace code for nets [i] based on digital (84, 103, 932066)-net over F16, using
- digital (2, 11, 10)-net over F4, using
(217−19, 217, large)-Net over F4 — Digital
Digital (198, 217, large)-net over F4, using
- t-expansion [i] based on digital (195, 217, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4217, large, F4, 22) (dual of [large, large−217, 23]-code), using
- strength reduction [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- strength reduction [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4217, large, F4, 22) (dual of [large, large−217, 23]-code), using
(217−19, 217, large)-Net in Base 4 — Upper bound on s
There is no (198, 217, large)-net in base 4, because
- 17 times m-reduction [i] would yield (198, 200, large)-net in base 4, but