---
title: Parallel Mixed Dimension Construction
url: https://www.emergentmind.com/topics/parallel-mixed-dimension-construction
type: topic
---

# Parallel Mixed Dimension Construction

Searching arXiv for the specified papers and closely related work on parallel and mixed-dimension constructions in constant-dimension coding.
Parallel mixed dimension construction is a method for building large constant dimension codes by combining two sources of codewords: one part comes from a mixed-dimension code on the left block of coordinates, and the other comes from a second mixed-dimension code on the right block, with a rank-metric “bridge” in between [2507.07842]. In the literature summarized here, the construction belongs to the broader development of parallel and linkage-type techniques for constant-dimension codes rather than to mixed-dimension coding in the strict sense. Its role is to assemble an \((n,2\delta,\{k\})_q\) constant dimension code from blockwise components while preserving subspace distance through MRD and RRMC constraints, and later work refines it through generalized bilateral multilevel methods and bilateral identifying vectors [2507.07842].

## 1. Position within constant-dimension code theory

A constant dimension code is a subset of the Grassmannian \(\mathcal G_q(n,k)\) with minimum subspace distance at least \(d\), where
\[
d_S(U,V)=\dim(U+V)-\dim(U\cap V).
\]
For fixed parameters, the central optimization problem is to maximize \(A_q(n,d,k)\), or in the notation used for the later bilateral framework, \(A_q(n,d,\{k\})\) [1910.04472][2507.07842].

The parallel mixed dimension construction is explicitly framed as a construction for constant dimension codes, not as a theory of mixed-dimension codes in the sense of allowing multiple output dimensions. The 2025 bilateral paper states that the method builds CDCs from mixed-dimension ingredients and then improves them by generalized bilateral multilevel construction [2507.07842]. Earlier related papers on linkage, generalized linkage, several parallel lifted MRD codes, and parallel multilevel constructions likewise remain focused on CDCs, even when they use “parallel” placements, rank-restricted components, or mixed ambient block sizes [1910.04472][1910.11195][1911.00154][1911.01878].

A recurring structural theme is the use of two or more coordinate orientations. In the 2019 linkage formulation, the key idea is to use both “parallel” orientations of linkage construction at the same time and then combine the resulting constant-dimension codewords while ensuring that the minimum subspace distance is preserved [1910.04472]. In the 2025 formulation, this theme appears in a more general blockwise form with mixed-dimension/distance subspace codes, MRD codes, RRMCs, and later bilateral multilevel fillings [2507.07842]. This suggests a historical continuity: the “parallel” aspect refers to simultaneous use of opposite block orientations, while the “mixed dimension” aspect refers to the auxiliary input codes rather than the final output family.

## 2. Foundational antecedents: linkage, lifted MRD codes, and parallelization

The immediate precursors are linkage-type and lifted-MRD-based constructions for CDCs. Standard linkage combines an existing CDC \(\mathcal U\) in one block with an MRD code in the remaining block to obtain a larger CDC. In the notation summarized for the 2019 linkage paper, if
\[
W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q\},
\]
then \(W_1\) is a CDC with parameters determined by the constituent CDC and MRD code [1910.04472]. The same paper observes that linkage can be done in two parallel ways,
\[
W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q_1\},
\qquad
W_2=\{\operatorname{Im}(Q\mid U): Q\in \mathcal Q_2,\ U\in \mathcal V\},
\]
and proves that the union \(W_1\cup W_2\) preserves the minimum subspace distance under a rank restriction on one side [1910.04472].

The several-parallel-lifted-MRD construction extends the same intuition by moving the identity block \(I_k\) through several positions and filling off-diagonal blocks by MRD codewords or subsets of MRD codewords with rank at most \(k-\frac d2\) [1911.00154]. The construction relies on Delsarte’s theorem to count low-rank codewords and yields new lower bounds for \(A_q((s+1)k+n,d,k)\) [1911.00154]. In parallel multilevel constructions, the intermediate object becomes a GRMC, which generalizes both ordinary rank-metric codes and constant-rank codes by specifying a rank set \(K\subseteq\{0,1,\dots,n\}\) [1911.01878]. There, the “parallel construction” already has the familiar two-family form
\[
C_1=\{\operatorname{rowspace}(I_k\mid A): A\in D_1\},
\qquad
C_2=\{\operatorname{rowspace}(B\mid I_k): B\in D_2\},
\]
with the rank bound on \(D_2\) ensuring the cross-distance condition [1911.01878].

The generalized linkage construction then unifies improved linkage and parallel linkage. Its central union
\[
\{T^{-1}(T(A)\mid M)\}\ \cup\ \{T^{-1}(R\mid T(B))\}
\]
combines CDCs, RMCs, and RRMCs and yields a lower bound that can strictly improve both earlier linkage variants [1910.11195]. Although that framework is not described as mixed-dimension coding, it already combines nonuniform block roles, lifted and non-lifted parts, and rank-restricted components. A plausible implication is that the later parallel mixed dimension construction inherits this blockwise combinatorial logic but replaces one-dimensional output families by more flexible mixed-dimension inputs [2507.07842].

## 3. Formal structure of the parallel mixed dimension construction

The 2025 bilateral paper identifies the parallel mixed dimension construction as a refinement of an earlier mixed dimension construction. A mixed dimension/distance subspace code (MDDC) is a code \(C\subseteq \mathcal P_q(n)\) with two distance parameters \(d_1\ge d_0\), such that equal-dimension pairs satisfy distance at least \(d_1\) and unequal-dimension pairs satisfy distance at least \(d_0\) [2507.07842].

The starting point is Theorem 2 from the earlier mixed dimension construction, which builds an \((n,2\delta,\{k\})_q\) CDC from two MDDCs on lengths \(n_1\) and \(n_2\), together with MRD and RRMC blocks, using matrices of the form
\[
\begin{pmatrix} H_1 & 0 \\ 0 & I_{k-t} \\ 0 & P \end{pmatrix},
\qquad
\begin{pmatrix} Q & I_{k-s} \\ H_2 & 0 \end{pmatrix},
\]
with appropriate rank constraints. The resulting CDC size is
\[
\sum_{t\in T_1} n_t(X_1)\,|P_t| + \sum_{s\in T_2} n_s(X_2)\,|Q_s| .
\]
Theorem 3, called the parallel mixed dimension construction, refines this by using one MDDC \(X_1\) on length \(n_1\) and another MDDC \(X_3\) on length \(n_3\), plus MRD and RRMC blocks, to build two families,
\[
C_1,\qquad C_3,
\]
which are then combined into an \((n,2\delta,\{k\})_q\) CDC,
\[
C_1\cup C_3,
\]
with size lower bound
\[
|C_1|+|C_3|
= \sum_{t\in T_1} n_t(X_1)\,|P_t|
+ \sum_{s\in T_2} n_s(X_3)\,|Q'_s| .
\]
A corollary simplifies the construction when \(n_3=k\) and \(T_2=\{k\}\) [2507.07842].

The construction is therefore “parallel” in the precise sense that it assembles two families arising from opposite sides of the coordinate decomposition. It is “mixed dimension” because the auxiliary codes \(X_1\) and \(X_3\) are mixed dimension/distance subspace codes. The output, however, remains constant dimension. This distinction matters because several related 2019 papers explicitly caution that their “parallel” constructions are not mixed-dimension code constructions in the strict sense [1910.04472][1910.11195].

## 4. Distance control, rank restrictions, and admissible block interactions

The decisive technical issue in all parallel constructions is the cross-distance between codewords coming from different families. In the two-sided linkage construction, the new difficulty is to control the distance between a word from \(W_1\) and a word from \(W_2\). The stated solution is a rank restriction on the second MRD-type code: each codeword has rank at most \(k-\frac d2\), which yields
\[
d(W_1,W_2)\ge 2k-2(k-\tfrac d2)=d
\]
[1910.04472].

The same mechanism appears in the lifted-MRD and GRMC settings. In the several-parallel-lifted-MRD construction, the off-diagonal matrices in later blocks are taken from the subset
\[
SQ_q(n,k,\tfrac d2)
\]
consisting of MRD codewords whose rank is at most \(k-\frac d2\), and the cardinality
\[
|SQ_q(n,k,\tfrac d2)|=\sum_{r=\frac d2}^{k-\frac d2} A_r\!\left(Q_q(n,k,\tfrac d2)\right)
\]
is computed via Delsarte’s theorem [1911.00154]. In the GRMC framework, the second lifted family uses a \((k\times(n-k),M,\delta,[0,k-\delta])_q\)-GRMC so that every off-diagonal matrix has rank at most \(k-\delta\), yielding a CDC with minimum subspace distance \(2\delta\) [1911.01878].

In generalized linkage, the RRMC factor is explicitly parameterized as \((k\times(r-t),\#R,d/2;k-d/2-t)_q\), and the lower bound
\[
A_q(r+s,d;k)\ge A_q(r,d;k)\,M(q,k,s,d/2)
+ A_q(s+t,d;k)\,A(q,k,r-t,d/2,k-d/2-t)
\]
shows how the RRMC term contributes to the second family [1910.11195].

The parallel mixed dimension construction uses the same pattern at a higher level of abstraction: MDDCs determine the left and right structural families, while MRD and RRMC blocks enforce the admissibility of the bridge matrices [2507.07842]. This suggests that the core invariant across the literature is not the particular encoding formalism but the rank-bounded control of intersections across differently oriented block placements.

## 5. Bilateral multilevel refinement of the construction

The 2025 paper’s main contribution is to refine the parallel mixed dimension construction through generalized bilateral multilevel construction [2507.07842]. This refinement introduces bilateral identifying vectors, inverse identifying vectors, generalized bilateral echelon Ferrers forms, and a rank-controlled upper-right submatrix.

A bilateral identifying vector has the form
\[
v=(v_1\mid v_3\mid v_2),
\]
where \(v_1\) is an identifying vector on the first \(n_1\) coordinates, \(v_2\) is an inverse identifying vector on the last \(n_2\) coordinates, and \(v_3\) is a zero block. If \(wt(v_1)=a_1\) and \(wt(v_2)=a_2\), then \(a_1+a_2=k\) [2507.07842]. The corresponding generalized bilateral echelon Ferrers form combines an echelon Ferrers form on the left, an inverse echelon Ferrers form on the right, a zero lower-left block, a constrained upper-right block, and a full Ferrers block in the middle coordinates [2507.07842].

Two distance lemmas support the bilateral method. If \(U,V\) correspond to bilateral identifying vectors \(u,v\) of the same type, then
\[
d_S(U,V)\ge d_H(u,v).
\]
If they correspond to the same bilateral identifying vector \(v\), then
\[
d_S(U,V)=2\,d_R(\check U,\check V),
\]
where \(\check U,\check V\) are the nonpivot submatrices remaining after removing the pivot columns [2507.07842].

Theorem 4 gives the compatibility criterion. Let \(S\) be a set of bilateral identifying vectors of length \(n\), weight \(k\), and minimum Hamming distance \(2\delta\), all of the same type. Assume that for every \(v=(v_1\mid v_3\mid v_2)\in S\),
\[
\delta \le wt(v_1)\le k-\delta.
\]
If for each \(v\in S\) there exists an \((F_v,\delta)_q\) GB-FD code \(C_v\) such that
\[
\operatorname{rank}(o(M))\le wt(v_1)-\delta
\quad \text{for all } M\in C_v,
\]
then
\[
C_1=\bigcup_{v\in S} L(C_v)
\]
is an \((n,2\delta,\{k\})_q\) CDC; moreover, if \(C_3\) is the CDC coming from the parallel mixed dimension construction, then
\[
C_1\cup C_3\cup \check C_1
\]
is also an \((n,2\delta,\{k\})_q\) CDC, with size
\[
|C_1|+|C_3|+|\check C_1|.
\]
Theorem 5 gives a simplified version when \(n_3=k\) and the right-hand component degenerates [2507.07842].

This refinement changes the construction at two levels. First, it replaces coarse pivot-pattern choices by bilateral identifying vectors subject to a Hamming-distance criterion. Second, it replaces generic lifted blocks by GB-FD codes with a rank restriction on the submatrix \(o(M)\). The paper explicitly states that the generalized bilateral multilevel construction makes each block denser while preserving the minimum distance [2507.07842].

## 6. Lower bounds, examples, and reported improvements

The literature consistently emphasizes that these constructions are intended to strengthen lower bounds for \(A_q(n,d,k)\) or \(A_q(n,d,\{k\})\). In the 2019 two-sided linkage paper, Theorem 2 produces an \((n_1+n_2,N_1N_2+N_3N_4,d,k)_q\) CDC, and the paper states improvements for
\[
A_q(13,4,4),\qquad A_q(17,4,4),\qquad A_q(19,6,6).
\]
For \(q=2\), the stated lower bounds are
\[
A_2(13,4,4)\ge 157337054,
\]
\[
A_2(17,4,4)\ge 644769570782,
\]
\[
A_2(19,6,6)\ge 4527333091203726
\]
[1910.04472].

The several-parallel-lifted-MRD paper gives a multi-parallel lower bound for \(A_q((s+1)k+n,d,k)\) and reports, for example, that when \(n=5\), \(k=5\), \(d=4\),
\[
A_q(15,4,5)\ge q^{40}+(A_2(Q_q(5,5,2))+A_3(Q_q(5,5,2)))q^{20}
+(A_2(Q_q(5,5,2))+A_3(Q_q(5,5,2)))^2.
\]
For \(q=2\), using
\[
A_2(Q_2(5,5,2))=4805,\qquad A_3(Q_2(5,5,2))=124930,
\]
the paper obtains
\[
A_2(15,4,5)\ge 1252379805361,
\]
larger than
\[
1235787711790
\]
[1911.00154].

The generalized linkage paper highlights the benchmark \(A_q(12,4;4)\). For \(q=2\), it reports an improvement from \(19{,}664{,}917\) to \(19{,}673{,}821\), while the parallel linkage construction gave only \(19{,}297{,}741\) [1910.11195]. The parallel multilevel paper states that the ratio between the new lower bound and the known upper bound for \((4\delta,2\delta,2\delta)_q\)-CDCs is greater than \(0.99926\) for any prime power \(q\) and any \(\delta\ge 3\) [1911.01878].

Within the specific parallel mixed dimension framework, the bilateral refinement gives explicit new values. In Example 6, with
\[
q=2,\ n=18,\ n_1=8,\ n_2=10,\ n_3=8,\ k=4,\ \delta=2,\ T_1=T_2=\{4,3\},
\]
the paper uses an \((8,4,3,\{4,3\})_2\) MDDC with
\[
|X_1|=4801,\qquad |X_3|=327,
\]
and obtains
\[
|C_1\cup C_3|
= 4801\cdot 2^{30} + 327\cdot 2^{27}
+ 4801\cdot A(4,10,2;2)_2 + 327\cdot A(4,9,2;2)_2
= 5199101447408.
\]
The new bilateral part contributes
\[
|\check C_1|=2413056,
\]
so that
\[
A_2(18,4,\{4\})\ge 5199103860464,
\]
improving the previous lower bound \(5199101447408\) [2507.07842].

In Example 7, for
\[
q=2,\ n=15,\ n_1=8,\ n_2=7,\ k=4,\ \delta=2,\ T_1=\{4,3\},
\]
the paper reports
\[
|C_1\cup C_3|=10154219486,\qquad |\check C_1|=44672,
\]
hence
\[
A_2(15,4,\{4\})\ge 10154264158,
\]
improving \(10154219486\) [2507.07842]. The same paper states that Table 1 gives at least 49 new lower bounds [2507.07842].

## 7. Interpretation, scope, and conceptual boundaries

The phrase “parallel mixed dimension construction” can be misleading if read outside the CDC context. The 2025 bilateral paper uses it for a construction whose inputs include mixed dimension/distance subspace codes but whose output is a constant dimension code [2507.07842]. Earlier related work makes the same distinction in other language. The 2019 linkage paper states that its construction does not primarily develop a mixed-dimension code construction and that the “parallel” language refers to two linkage orientations, not to mixed-dimension codes in the strict sense [1910.04472]. The generalized linkage paper similarly states that it is fundamentally about constant-dimension codes, though it incorporates “mixed” behavior through different ambient block sizes and the joint use of CDCs and RRMCs [1910.11195].

Within this boundary, the method has a clear technical identity. It combines blockwise orientations, rank-metric bridges, and rank restrictions so that multiple families of lifted or partially lifted subspaces can coexist in one CDC. In its later bilateral form, it further combines these ingredients with identifying-vector selection and Ferrers-diagram fillers under explicit rank constraints on a designated submatrix [2507.07842]. A plausible implication is that the construction is best understood not as a departure from the CDC framework but as a synthesis of three earlier lines: two-sided linkage, several parallel lifted MRD constructions, and multilevel/Ferrers-based refinement [1910.04472][1911.00154][1911.01878].

Its scope is also parameter-sensitive. The two-sided linkage paper notes that the main construction in Theorems 2 and 3 is most directly applicable when
\[
d\le k,
\]
and that for \(d>k\) it uses rank-restricted rank-metric codes instead [1910.04472]. The generalized linkage framework introduces the parameter \(t\) to interpolate between improved linkage and parallel linkage, and its flexibility can strictly improve both older constructions for infinite parameter families [1910.11195]. The bilateral refinement then densifies the parallel mixed dimension skeleton by inserting additional codewords \(\check C_1\) while keeping the distance conditions intact [2507.07842].

In summary, parallel mixed dimension construction denotes a CDC construction paradigm in which two oppositely oriented block families, derived from mixed-dimension inputs and connected by MRD or RRMC bridge components, are combined into a single constant dimension code. Its later bilateral generalization adds compatible bilateral identifying vectors and GB-FD fillers, thereby enlarging the code size and improving many previously best-known lower bounds [2507.07842].

Source: https://www.emergentmind.com/topics/parallel-mixed-dimension-construction