Best Known (229−23, 229, s)-Nets in Base 4
(229−23, 229, 762648)-Net over F4 — Constructive and digital
Digital (206, 229, 762648)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (13, 24, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 12, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- trace code for nets [i] based on digital (1, 12, 24)-net over F16, using
- digital (182, 205, 762600)-net over F4, using
- net defined by OOA [i] based on linear OOA(4205, 762600, F4, 23, 23) (dual of [(762600, 23), 17539595, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4205, 8388601, F4, 23) (dual of [8388601, 8388396, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4205, 8388601, F4, 23) (dual of [8388601, 8388396, 24]-code), using
- net defined by OOA [i] based on linear OOA(4205, 762600, F4, 23, 23) (dual of [(762600, 23), 17539595, 24]-NRT-code), using
- digital (13, 24, 48)-net over F4, using
(229−23, 229, large)-Net over F4 — Digital
Digital (206, 229, large)-net over F4, using
- 42 times duplication [i] based on digital (204, 227, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4227, large, F4, 23) (dual of [large, large−227, 24]-code), using
- 22 times code embedding in larger space [i] based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- 22 times code embedding in larger space [i] based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4227, large, F4, 23) (dual of [large, large−227, 24]-code), using
(229−23, 229, large)-Net in Base 4 — Upper bound on s
There is no (206, 229, large)-net in base 4, because
- 21 times m-reduction [i] would yield (206, 208, large)-net in base 4, but