Abstract
A basic problem of the constant dimension subspace coding is to determine the maximal possible size Aq(n,d,k) of a set of k-dimensional subspaces in \(\mathbf {F}_{q}^{n}\) such that the subspace distance satisfies \(\text {dis}(U,V) =2k-2 \dim (U \cap V) \geq d\) for any two different subspaces U and V in this set. We propose two constructions of constant dimension subspace codes that can insert flexibly into the generalized parallel linkage construction. In our constructions matrix blocks from small constant dimension codes and rank metric codes play important roles. Through a well-arranged combination for the matrix blocks, more than 120 new constant dimension subspace codes of distance 4, 6, 8 better than previously best known codes are constructed.
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Notes
With the same notation l in Lemma 4.4 of [2], the construction is rewritten for the case of l = 2.
The values of the column named “New” are the new lower bounds of CDCs from our constructions. The values of the column named “Old” are the new lower bounds of CDCs from [12].
Table 1 New lower bounds of q = 2
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Acknowledgements
The research of Hao Chen was supported by National Natural Science Foundation of China (NSFC) Grant 62032009. The research of Xiaoqing Tan was supported by NSFC Grant 61672014, National Cryptography Development Fund of China Grant MMJJ20180109, and Natural Science Foundation of Guangdong Province of China Grant 2019A1515011069. This research was supported by the Major Program of Guangdong Basic and Applied Research under Grant 2019030302008.
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Appendix: A
Appendix: A
In [10], the authors proposed a construction that composes several CDCs in Theorem 4. However, the result is incorrect since the minimal distance of the CDCs in construction is not satisfied ≥ d under some conditions.
More specifically, the proof of dis(W3,W4) ≥ d in Theorem 4 is reduced to \(\dim (W_{3} + W_{4}) \geq k + \frac {d}{2}\), where W3 is a subspace in the CDC B1 and W4 is a subspace in the CDC B2. The authors claimed that \(\dim (W_{3} + W_{4}) = \dim (W_{3} + W_{4}^{\prime }) = 2k - \dim (W_{3} \cap W_{4}^{\prime }) \geq k + \frac {d}{2}\), where \(W_{4}^{\prime } = \text {rs} \left (\begin {array}{lllllll} I_{a_{2}} & | & M_{2} & | & \quad O_{2} & \\ O_{1} & |& M_{3} & | & O^{\prime } & | & M_{1}^{\prime } \end {array}\right )\), M3 is a matrix with restricted rank \(\leq a_{1} - \frac {d}{2}\) and \(M_{1}^{\prime }\) is a matrix with unknown rank.
The problem is that \(\dim (W_{3} + W_{4}^{\prime }) = 2k - \dim (W_{3} \cap W_{4}^{\prime })\) is not strictly proved in the paper. It is true if and only if \(\dim (W_{3}) = \dim (W_{4}^{\prime })=k\). However, the Theorem 4 only gives that \(\dim (W_{4}^{\prime }) = a_{2} + \text {rank} (M_{3} | O^{\prime } | M_{1}^{\prime }) \leq a_{2} + \text {rank} (M_{3}) + \text {rank} (M_{1}^{\prime }) \leq k - \frac {d}{2} + \text {rank} (M_{1}^{\prime })\). Thus \(\dim (W_{4}^{\prime })=k\) is not satisfied if \(\text {rank} (M_{3} | O^{\prime } | M_{1}^{\prime }) \neq a_{1}\).
We give a concrete example to show that the minimal distance is not satisfied in some cases. Using the same notations in Theorem 4 of [10], let n1 = n2 = 6,a1 = 4,a2 = 2,b1 = b2 = 2,d = 4,k = 6 and q = 2, then M1 ∈ (4, 2, 2)2 RMC, M2 ∈ (2, 4, 2)2 RMC and M3 ∈ (4, 4, 2)2 RRMC with rank ≤ 2. For simplification, we further assume M1 and M2 are zero matrices, and \(M_{3} = \left (\begin {array}{llll} {1} {0} {0} {0} \\ {0} {0} {0} {0} \\ {0} {0} {0} {0} \\ {0} {1} {0} {0} \end {array}\right )\). From the construction, we have
Using the same proof technique, \( \dim (W_{3} + W_{4}) = \text {rank} \left (\begin {array}{ll} G_{3} \\ G_{4} \end {array}\right ) = \text {rank} \left (\begin {array}{ll} G_{3} \\ G_{4}^{\prime } \end {array}\right ) = \dim (W_{3} + W_{4}^{\prime }), \) where \(W_{4}^{\prime } = \text {rs} (G_{4}^{\prime }) = \text {rs} \left (\begin {array}{llllllllllll} 1 0 0 0 0 0 | 0 0 0 0 0 0 \\ 0 1 0 0 0 0 | 0 0 0 0 0 0 \\ 0 0 1 0 0 0 | {0} 0 0 0 0 0 \\ 0 0 0 0 0 0 | 0 {0} 0 0 0 0 \\ 0 0 0 0 0 0 | 0 0 1 0 0 0 \\ 0 0 0 1 0 0 | 0 0 0 1 0 0 \end {array}\right )\) and the first (second) green zero is the result of subtracting the 9th (10th) row from the 5th (6th) row of \(\left (\begin {array}{ll} G_{3} \\ G_{4} \end {array}\right )\).
It is easy to check that \(\dim (W_{3}) = \dim (W_{4}) = 6\). According to the Theorem 4, dis(W3,W4) ≥ 4 should be satisfied, which implies that \(\dim (W_{3} + W_{4})\) should be ≥ 8. However, \(\dim (W_{4}^{\prime }) = 5\) and
Thus \(\text {dis}(W_{3}, W_{4}) = 2\dim (W_{3} + W_{4}) - 2 \times 6 = 14 - 12 = 2\), which not meets the desired minimal distance.
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Lao, H., Chen, H. & Tan, X. New constant dimension subspace codes from block inserting constructions. Cryptogr. Commun. 14, 87–99 (2022). https://doi.org/10.1007/s12095-021-00524-9
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DOI: https://doi.org/10.1007/s12095-021-00524-9