Best Known (55, 55+8, s)-Nets in Base 4
(55, 55+8, 262149)-Net over F4 — Constructive and digital
Digital (55, 63, 262149)-net over F4, using
- 1 times m-reduction [i] based on digital (55, 64, 262149)-net over F4, using
- net defined by OOA [i] based on linear OOA(464, 262149, F4, 9, 9) (dual of [(262149, 9), 2359277, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(464, 1048597, F4, 9) (dual of [1048597, 1048533, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [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(441, 1048576, F4, 6) (dual of [1048576, 1048535, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(464, 1048597, F4, 9) (dual of [1048597, 1048533, 10]-code), using
- net defined by OOA [i] based on linear OOA(464, 262149, F4, 9, 9) (dual of [(262149, 9), 2359277, 10]-NRT-code), using
(55, 55+8, 1048600)-Net over F4 — Digital
Digital (55, 63, 1048600)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(463, 1048600, F4, 8) (dual of [1048600, 1048537, 9]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(462, 1048598, F4, 8) (dual of [1048598, 1048536, 9]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(5) [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(441, 1048576, F4, 6) (dual of [1048576, 1048535, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(421, 22, F4, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,4)), using
- dual of repetition code with length 22 [i]
- linear OA(41, 22, F4, 1) (dual of [22, 21, 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(5) [i] based on
- linear OA(462, 1048599, F4, 7) (dual of [1048599, 1048537, 8]-code), using Gilbert–Varšamov bound and bm = 462 > Vbs−1(k−1) = 1 345996 089003 884842 401106 373357 163064 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(462, 1048598, F4, 8) (dual of [1048598, 1048536, 9]-code), using
- construction X with Varšamov bound [i] based on
(55, 55+8, large)-Net in Base 4 — Upper bound on s
There is no (55, 63, large)-net in base 4, because
- 6 times m-reduction [i] would yield (55, 57, large)-net in base 4, but