Best Known (240, 240+17, s)-Nets in Base 4
(240, 240+17, 4456446)-Net over F4 — Constructive and digital
Digital (240, 257, 4456446)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (53, 61, 262146)-net over F4, using
- net defined by OOA [i] based on linear OOA(461, 262146, F4, 8, 8) (dual of [(262146, 8), 2097107, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(461, 1048584, F4, 8) (dual of [1048584, 1048523, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(461, 1048587, F4, 8) (dual of [1048587, 1048526, 9]-code), using
- 1 times truncation [i] based on linear OA(462, 1048588, F4, 9) (dual of [1048588, 1048526, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(461, 1048576, F4, 9) (dual of [1048576, 1048515, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(451, 1048576, F4, 7) (dual of [1048576, 1048525, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(411, 12, F4, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,4)), using
- dual of repetition code with length 12 [i]
- linear OA(41, 12, F4, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 1 times truncation [i] based on linear OA(462, 1048588, F4, 9) (dual of [1048588, 1048526, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(461, 1048587, F4, 8) (dual of [1048587, 1048526, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(461, 1048584, F4, 8) (dual of [1048584, 1048523, 9]-code), using
- net defined by OOA [i] based on linear OOA(461, 262146, F4, 8, 8) (dual of [(262146, 8), 2097107, 9]-NRT-code), using
- digital (179, 196, 4194300)-net over F4, using
- trace code for nets [i] based on digital (81, 98, 2097150)-net over F16, using
- net defined by OOA [i] based on linear OOA(1698, 2097150, F16, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1698, 8388601, F16, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1698, 8388602, F16, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
- trace code [i] based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- trace code [i] based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1698, 8388602, F16, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1698, 8388601, F16, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1698, 2097150, F16, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
- trace code for nets [i] based on digital (81, 98, 2097150)-net over F16, using
- digital (53, 61, 262146)-net over F4, using
(240, 240+17, large)-Net over F4 — Digital
Digital (240, 257, large)-net over F4, using
- t-expansion [i] based on digital (231, 257, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
- 28 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- 28 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
(240, 240+17, large)-Net in Base 4 — Upper bound on s
There is no (240, 257, large)-net in base 4, because
- 15 times m-reduction [i] would yield (240, 242, large)-net in base 4, but