Best Known (256−33, 256, s)-Nets in Base 4
(256−33, 256, 65540)-Net over F4 — Constructive and digital
Digital (223, 256, 65540)-net over F4, using
- net defined by OOA [i] based on linear OOA(4256, 65540, F4, 33, 33) (dual of [(65540, 33), 2162564, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4256, 1048641, F4, 33) (dual of [1048641, 1048385, 34]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4254, 1048639, F4, 33) (dual of [1048639, 1048385, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(25) [i] based on
- linear OA(4241, 1048576, F4, 33) (dual of [1048576, 1048335, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4191, 1048576, F4, 26) (dual of [1048576, 1048385, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- construction X applied to Ce(32) ⊂ Ce(25) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4254, 1048639, F4, 33) (dual of [1048639, 1048385, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4256, 1048641, F4, 33) (dual of [1048641, 1048385, 34]-code), using
(256−33, 256, 502444)-Net over F4 — Digital
Digital (223, 256, 502444)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4256, 502444, F4, 2, 33) (dual of [(502444, 2), 1004632, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4256, 524320, F4, 2, 33) (dual of [(524320, 2), 1048384, 34]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4255, 524320, F4, 2, 33) (dual of [(524320, 2), 1048385, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4255, 1048640, F4, 33) (dual of [1048640, 1048385, 34]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4254, 1048639, F4, 33) (dual of [1048639, 1048385, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(25) [i] based on
- linear OA(4241, 1048576, F4, 33) (dual of [1048576, 1048335, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4191, 1048576, F4, 26) (dual of [1048576, 1048385, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- construction X applied to Ce(32) ⊂ Ce(25) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4254, 1048639, F4, 33) (dual of [1048639, 1048385, 34]-code), using
- OOA 2-folding [i] based on linear OA(4255, 1048640, F4, 33) (dual of [1048640, 1048385, 34]-code), using
- 41 times duplication [i] based on linear OOA(4255, 524320, F4, 2, 33) (dual of [(524320, 2), 1048385, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4256, 524320, F4, 2, 33) (dual of [(524320, 2), 1048384, 34]-NRT-code), using
(256−33, 256, large)-Net in Base 4 — Upper bound on s
There is no (223, 256, large)-net in base 4, because
- 31 times m-reduction [i] would yield (223, 225, large)-net in base 4, but