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New Constructions of Subspace Codes Using Subsets of MRD codes in Several Blocks

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Pith's one-line read Two constructions that pack bounded-rank slices of MRD codes into parallel blocks prove more than 110 new lower bounds for constant-dimension subspace codes, each improving the best previously known value.

desk verdict Sound parallel-block constructions with real lower-bound improvements; the math holds up, and only the presentation needs tightening. read the letter →

arxiv 1908.03804 v2 pith:ULPC3S74 submitted 2019-08-10 cs.IT math.IT

Hao Chen , Xianmang He , Jian Weng , Liqing Xu This is my paper · ORCID
classification cs.ITmath.IT MSC 94B2594B6511T71
keywords constantdimensionsubspacecodesrank-metricMRDbounded-ranksubsetsrankdistributionlinkageconstructionliftedlowerbounds
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper targets the core packing question of constant-dimension subspace coding: how many $k$-dimensional subspaces of $\mathbb{F}_q^n$ can be placed so that any two are at subspace distance at least $d$? It presents two constructions that produce lower bounds for this maximum, $A_q(n,d,k)$, and claims that more than 110 of the resulting bounds are strictly better than the best previously known values. The first construction is a parallel form of the linkage method; the second glues together arbitrary numbers of lifted maximum rank-distance (MRD) codes. In both, the crucial new ingredient is the use of subsets of MRD codes consisting only of matrices of bounded rank, counted exactly by the rank-distribution theorem for MRD codes. If the claims are right, the standard tables of best known subspace-code sizes can be improved in these parameter ranges.

What carries the argument

The central objects are the bounded-rank subsets $Q_{q,n,t,k} = \{f \in Q_{q,n,t} : \dim \ker f \ge k\}$ of the MRD code of $q$-polynomial linear maps on $\mathbb{F}_{q^n}$ with rank distance $n-t$; their sizes are computed from the rank distribution $A_i(Q_{q,n,t})$ of the MRD code. The identity that carries the argument is the cross-block intersection bound of Proposition 4.1: for two lifted row spaces whose identity blocks are in positions $i<j$, the intersection has dimension at most $n - \mathrm{rank}(I_n - A_j B_i)$, and because the matrix $B_i$ sitting in the earlier block position has rank at most $t$, this dimension is at most $t$. This single estimate turns the disjoint union of parallel lifted MRD codes into a valid constant-dimension subspace code, and the rank-distribution sums then count the codewords.

What would settle it

Take a concrete case such as $q=2$, $n=6$, $t=3$, $s=2$ and explicitly compute the intersection of two lifted subspaces from different block positions; any pair with intersection dimension 4 or more refutes the claimed distance $2(n-t)=6$. An independent exact determination of $A_2(18,6,6)$ below the claimed lower bound would also settle the claim.

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Extended reading notes

Core claim

The paper's central claim is that bounded-rank subsets of MRD codes can be packed into several parallel blocks without losing distance. Theorem 4.1 states that whenever $2t \ge n$, for every $s \ge 1$, $A_q((s+1)n, 2(n-t), n) \ge \sum_{j=0}^{s} q^{(s-j)n(t+1)} (\sum_{i=n-t}^{t} A_i(Q_{q,n,t}))^j$, where $A_i(Q_{q,n,t})$ is the number of rank-$i$ matrices in the $q$-polynomial MRD code $Q_{q,n,t}$. The construction places the $n \times n$ identity block in each of the $s+1$ positions; for a subspace whose identity lies in block $j$, the matrices in blocks left of the identity are required to have kernel dimension at least $n-t$, i.e., rank at most $t$. That rank restriction makes the intersection of any two subspaces from different block positions have dimension at most $t$, so the subspace distance is at least $2(n-t)$. A parallel linkage construction (Theorem 3.1) gives a similar bound for parameters $(3k+h, d, k)$ with even $d$, and together the two methods yield more than 110 lower bounds better than those in [11].

Load-bearing premise

The construction's distance guarantee rests entirely on the estimate that a rank-at-most-$t$ matrix in an earlier block keeps the intersection of two lifted subspaces to dimension at most $t$; if that estimate ever failed, the minimum subspace distance $2(n-t)$ would drop and the claimed code would not exist.

Editorial extensions

If this is right

  • The $s=1$ case of Theorem 4.1 gives $A_q(2n, 2(n-t), n) \ge q^{n(t+1)} + \sum_{i=n-t}^{t} A_i(Q_{q,n,t})$, producing 42 entries in the paper's tables that improve previous records.
  • Applying the Johnson-type bound to the $s=1$ construction yields 7 improved lower bounds for parameters of the form $A_q(2n-1, 2(n-t), n-1)$, including $A_2(17,6,8)$.
  • The parallel linkage construction of Theorem 3.1 and Corollary 3.1 gives 63 improved bounds for $A_q(3k+h, d, k)$ with even $d$, such as $A_2(18,6,6)$ and $A_2(19,6,6)$.
  • The $s=2$ and $s=3$ cases of the multi-block construction produce new lower bounds, including values for $A_2(20,4,5)$ and $A_2(24,6,6)$ that cannot be compared with any entry in [11].
  • In total, the two methods yield more than 110 constant-dimension subspace codes whose sizes exceed the best previously known lower bounds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same mechanism should work with blocks of different sizes or with different MRD codes in different blocks, because the cross-block distance estimate uses only the rank bound on one matrix.
  • For fixed $n$ and $t$, the relative gain over a single lifted MRD code is largest at small $s$; as $s$ grows, the highest-power term in the sum dominates and the construction approaches the behaviour of one large block.
  • Because the bounded-rank counts are exact polynomials in $q$, the new bounds can be plugged into the anticode upper bound to measure, for each $q$, how close these parameters come to the theoretical maximum.
  • A natural testable extension is to relax the condition $2t \ge n$ by replacing the single low-rank restriction with a two-sided rank restriction, which might allow shorter code lengths.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper studies lower bounds for the maximal size A_q(n,d,k) of constant-dimension subspace codes. It proposes two constructions. The first (Theorem 3.1, Corollary 3.1) is a parallel linkage construction: two families of linked or lifted codes are combined, with one family using rank-restricted subsets of an MRD code as the linking block, and the Delsarte rank distribution is used to count the restricted subset. The second (Theorem 4.1, Corollaries 4.1–4.6) uses s+1 blocks and takes subspaces spanned by rows (A_1,...,I_n,...,A_s), with the identity block in each of the s+1 positions; cross-block intersections are controlled by imposing that the blocks to the left of the identity have rank at most t, and the count is expressed as a sum over the rank distribution of the MRD code Q_{q,n,t}. The constructions yield explicit lower bounds, and the appendix lists many entries claimed to improve the tables in [11].

Significance. The main mathematical content is sound: the distance bound in Theorem 4.1 is valid once the notation is made precise, and the counting formula in that theorem is consistent with the construction. The paper's strength is the systematic use of rank-bounded subsets of MRD codes in several blocks together with the Delsarte rank distribution, which is a clean and reusable idea. The Delsarte-based counting is explicit and can be checked directly from the stated formulas. If the presentation is completed (full tables, resolved placeholders, updated comparison with later work), the paper provides a substantial number of new lower bounds and will be a useful reference for the subspace-codes community.

major comments (3)
  1. [§4.1 (Proposition 4.1 and Theorem 4.1)] The notation for the block entries is not defined consistently: the proposition is stated with A_j and B_i, while the proofs of Theorem 4.1 switch to expressions such as A_i^j A_j^i and A_j^i without saying which superscript is the block position and which is the identity position. The load-bearing step is the bound dim(U^i ∩ U^j) ≤ t; as written, the statement that dim ker(I_n − A_i^j A_j^i) ≤ t follows from dim ker(A_j^i) ≥ n−t requires the reader to reconstruct which matrix is the constrained one and to use rank(AB) ≤ rank(B). Proposition 4.1 as stated also asserts the bound for arbitrary A_j and B_i, but the bound to t is true only under the rank constraint on the appropriate block entry. Please define the notation, state the constraint explicitly, and expand the argument, because the minimum distance 2(n−t) of Theorem 4.1 rests entirely on this step.
  2. [§4.2, §5, and Appendix] The manuscript repeatedly refers to 'the full version [ ?]' (for example, §4.2: 'We list all 42 improvements on [11] in Table 2 of the full version [ ?]', and §5: 'Tables 1-5 in the full version [ ?]'). The submitted version contains an appendix with tables, but the unresolved placeholder and the referral to an unidentifiable full version make the claimed 'more than 110 new lower bounds' impossible to verify from the manuscript. Please remove the placeholder, include the complete set of tables or a permanent archive link, and ensure that the count of 110 is supported by the table entries.
  3. [§5 (Added Notes)] The paper compares all new bounds with the 2016 tables [11], but the Added Notes already acknowledge that some results have been extended in [3,13,19]. The manuscript should either update the comparison to the current best-known bounds or explicitly state which of the 110 listed entries remain improvements in the presence of the later work. Without this, the claim that these are 'new constant dimension subspace codes better than previously best known codes' is not substantiated at the time of publication.
minor comments (5)
  1. [§2.2] The sentence 'We refer the following result to Theorem 5.6 in [5] or Corollary 26 in [4]' should read 'We refer to the following result'.
  2. [§4.1] In the paragraph introducing the subspaces U^j, 'B1,...,Bn are matrices from the MRD code' should be 'B1,...,B_s are matrices from the MRD code'.
  3. [§4.2] The sentence 'From Theorem 4.1 we have the following Corollary 3.2 immediately' should refer to Corollary 4.3, not Corollary 3.2.
  4. [Appendix, Table 4] The table heading 'A_q(3n. 2(n−t), n)' contains a period where a comma is intended; it should read 'A_q(3n, 2(n−t), n)'.
  5. [Appendix] The large integers in the tables are broken across lines without alignment; consider presenting them in a monospaced or comma-separated format so that each entry can be checked reliably against the formulas.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the central constructions and counts follow from explicit intersection arguments and the external Delsarte rank-distribution theorem.

full rationale

The main derivation chain is self-contained rather than circular. Theorem 4.1 is proved by an explicit construction: for each block position of the identity, the entries before the identity are restricted to the subset Q_{q,n,t,n-t}, whose size is counted by the Delsarte rank-distribution formula, and the remaining entries range over the full MRD code Q_{q,n,t}. The resulting count matches the construction term-by-term, and the minimum-distance claim rests on Proposition 4.1, which is proved directly from the rank of the relevant matrices and the kernel-dimension condition dim(ker(A_j^i)) >= n-t. This is a mathematical argument from stated assumptions, not a restatement of the target bound. Theorem 3.1 is likewise proved by a direct intersection calculation using the rank bound on Q_2; it does not import the claimed lower bound. The only self-citation that appears is the parenthetical reference to [26] for the value A_2(12,6,6) >= 16865101 in Section 3.2, and that same value is independently proved in this paper as Corollary 4.1 from the Delsarte theorem, so the self-citation is not load-bearing. No fitted input is renamed as a prediction, no uniqueness theorem from the authors is invoked to forbid alternatives, and no external benchmark is used circularly. The unresolved placeholder 'full version [ ?]' and the reliance on [11] for comparison affect exposition and table completeness, but they do not make any derivation equivalent to its own inputs. Therefore the paper receives a circularity score of 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard MRD code theory, the Delsarte rank distribution, and standard bounds from the cited literature. There are no fitted constants, no new postulated objects, and no ad hoc model assumptions beyond the explicit parameter restrictions (for example, 2t ≥ n and d even) that are stated in the theorems. The only novel objects are the parallel block constructions themselves, which are defined directly from existing MRD codes.

assumptions (4)
  • standard math Delsarte's rank distribution theorem for MRD codes
    Used in Theorem 2.1 and throughout to count |Q_{q,n,t,n-t}| = Σ_{i=n-t}^{t} A_i(Q_{q,n,t}); cited to [5] and [4] and not proved in the paper.
  • standard math Gabidulin MRD codes with rank distance n-t exist with q^{n(t+1)} elements
    Basis of the lifted MRD construction and of the subsets Q_{q,n,t}; cited to [9,5,22].
  • standard math A nonzero q-polynomial of q-degree t has at most q^t roots in F_{q^n}
    Gives the rank distance n-t for Q_{q,n,t} and bounds kernel dimensions; used in §2.1 and §4.1.
  • standard math Johnson type bound A_q(2n-1, 2(n-t), n-1) ≥ (1/(q^n+1)) A_q(2n, 2(n-t), n)
    External lower bound used in Corollary 4.2 to derive additional new codes; cited to [8].

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Pith. "Pith review of New Constructions of Subspace Codes Using Subsets of MRD codes in Several Blocks." pith.science (2026). https://pith.science/paper/ULPC3S74

@misc{pith2026190803804,
  author       = {Pith},
  title        = {Pith review of: New Constructions of Subspace Codes Using Subsets of MRD codes in Several Blocks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ULPC3S74}},
  note         = {Machine review of arXiv:1908.03804}
}
read the original abstract

A basic problem for the constant dimension subspace coding is to determine the maximal possible size A_q (n, d, k) of a set of k-dimensional subspaces in Fnq such that the subspace distance satisfies d(U, V )> or =d for any two different subspaces U andV in this set. We present two new constructions of constant dimension subspace codes using subsets of maximal rank-distance (MRD) codes in several blocks. This method is firstly applied to the linkage construction and secondly to arbitrary number of blocks of lifting MRD codes. In these two constructions, subsets of MRD codes with bounded ranks play an essential role. The Delsarte theorem of the rank distribution of MRD codes is an important ingredient to count codewords in our constructed constant dimension subspace codes. We give many new lower bounds for A_q (n, d, k). More than 110 new constant dimension subspace codes better than previously best known codes are constructed.

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