Parallel Mixed Dimension Construction

• Parallel mixed dimension construction is a technique for building constant dimension codes by merging two mixed-dimension code families via a rank-metric bridge.
• The method refines earlier linkage and lifted MRD constructions by using complementary block orientations and strict rank restrictions to maintain subspace distance.
• Recent bilateral multilevel refinements enhance code size and improve lower bounds by integrating identifying vectors with tailored Ferrers diagram fillers.

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 (Li et al., 10 Jul 2025). 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 (Li et al., 10 Jul 2025).

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, $A_q(n,d,\{k\})$ (He, 2019, Li et al., 10 Jul 2025).

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 (Li et al., 10 Jul 2025). 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 (He, 2019, Heinlein, 2019, He et al., 2019, Liu et al., 2019).

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 (He, 2019). 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 (Li et al., 10 Jul 2025). 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 (He, 2019). 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 $\mathcal G_q(n,k)$0 preserves the minimum subspace distance under a rank restriction on one side (He, 2019).

The several-parallel-lifted-MRD construction extends the same intuition by moving the identity block $\mathcal G_q(n,k)$1 through several positions and filling off-diagonal blocks by MRD codewords or subsets of MRD codewords with rank at most $\mathcal G_q(n,k)$2 (He et al., 2019). The construction relies on Delsarte’s theorem to count low-rank codewords and yields new lower bounds for $\mathcal G_q(n,k)$3 (He et al., 2019). 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 $\mathcal G_q(n,k)$4 (Liu et al., 2019). There, the “parallel construction” already has the familiar two-family form

$\mathcal G_q(n,k)$5

with the rank bound on $\mathcal G_q(n,k)$6 ensuring the cross-distance condition (Liu et al., 2019).

The generalized linkage construction then unifies improved linkage and parallel linkage. Its central union

$\mathcal G_q(n,k)$7

combines CDCs, RMCs, and RRMCs and yields a lower bound that can strictly improve both earlier linkage variants (Heinlein, 2019). 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 (Li et al., 10 Jul 2025).

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 $\mathcal G_q(n,k)$8 with two distance parameters $\mathcal G_q(n,k)$9, such that equal-dimension pairs satisfy distance at least $d$0 and unequal-dimension pairs satisfy distance at least $d$1 (Li et al., 10 Jul 2025).

The starting point is Theorem 2 from the earlier mixed dimension construction, which builds an $d$2 CDC from two MDDCs on lengths $d$3 and $d$4, together with MRD and RRMC blocks, using matrices of the form

$d$5

with appropriate rank constraints. The resulting CDC size is

$d$6

Theorem 3, called the parallel mixed dimension construction, refines this by using one MDDC $d$7 on length $d$8 and another MDDC $d$9 on length $d_S(U,V)=\dim(U+V)-\dim(U\cap V).$0, plus MRD and RRMC blocks, to build two families,

$d_S(U,V)=\dim(U+V)-\dim(U\cap V).$1

which are then combined into an $d_S(U,V)=\dim(U+V)-\dim(U\cap V).$2 CDC,

$d_S(U,V)=\dim(U+V)-\dim(U\cap V).$3

with size lower bound

$d_S(U,V)=\dim(U+V)-\dim(U\cap V).$4

A corollary simplifies the construction when $d_S(U,V)=\dim(U+V)-\dim(U\cap V).$5 and $d_S(U,V)=\dim(U+V)-\dim(U\cap V).$6 (Li et al., 10 Jul 2025).

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 $d_S(U,V)=\dim(U+V)-\dim(U\cap V).$7 and $d_S(U,V)=\dim(U+V)-\dim(U\cap V).$8 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 (He, 2019, Heinlein, 2019).

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 $d_S(U,V)=\dim(U+V)-\dim(U\cap V).$9 and a word from $A_q(n,d,k)$0. The stated solution is a rank restriction on the second MRD-type code: each codeword has rank at most $A_q(n,d,k)$1, which yields

$A_q(n,d,k)$2

(He, 2019).

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

$A_q(n,d,k)$3

consisting of MRD codewords whose rank is at most $A_q(n,d,k)$4, and the cardinality

$A_q(n,d,k)$5

is computed via Delsarte’s theorem (He et al., 2019). In the GRMC framework, the second lifted family uses a $A_q(n,d,k)$6-GRMC so that every off-diagonal matrix has rank at most $A_q(n,d,k)$7, yielding a CDC with minimum subspace distance $A_q(n,d,k)$8 (Liu et al., 2019).

In generalized linkage, the RRMC factor is explicitly parameterized as $A_q(n,d,k)$9, and the lower bound

$A_q(n,d,\{k\})$0

shows how the RRMC term contributes to the second family (Heinlein, 2019).

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 (Li et al., 10 Jul 2025). 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 (Li et al., 10 Jul 2025). 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

$A_q(n,d,\{k\})$1

where $A_q(n,d,\{k\})$2 is an identifying vector on the first $A_q(n,d,\{k\})$3 coordinates, $A_q(n,d,\{k\})$4 is an inverse identifying vector on the last $A_q(n,d,\{k\})$5 coordinates, and $A_q(n,d,\{k\})$6 is a zero block. If $A_q(n,d,\{k\})$7 and $A_q(n,d,\{k\})$8, then $A_q(n,d,\{k\})$9 (Li et al., 10 Jul 2025). 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 (Li et al., 10 Jul 2025).

Two distance lemmas support the bilateral method. If $\mathcal U$0 correspond to bilateral identifying vectors $\mathcal U$1 of the same type, then

$\mathcal U$2

If they correspond to the same bilateral identifying vector $\mathcal U$3, then

$\mathcal U$4

where $\mathcal U$5 are the nonpivot submatrices remaining after removing the pivot columns (Li et al., 10 Jul 2025).

Theorem 4 gives the compatibility criterion. Let $\mathcal U$6 be a set of bilateral identifying vectors of length $\mathcal U$7, weight $\mathcal U$8, and minimum Hamming distance $\mathcal U$9, all of the same type. Assume that for every $$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q\},$$0,

$$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q\},$$1

If for each $$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q\},$$2 there exists an $$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q\},$$3 GB-FD code $$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q\},$$4 such that

$$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q\},$$5

then

$$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q\},$$6

is an $$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q\},$$7 CDC; moreover, if $$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q\},$$8 is the CDC coming from the parallel mixed dimension construction, then

$$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q\},$$9

is also an $W_1$0 CDC, with size

$W_1$1

Theorem 5 gives a simplified version when $W_1$2 and the right-hand component degenerates (Li et al., 10 Jul 2025).

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 $W_1$3. The paper explicitly states that the generalized bilateral multilevel construction makes each block denser while preserving the minimum distance (Li et al., 10 Jul 2025).

6. Lower bounds, examples, and reported improvements

The literature consistently emphasizes that these constructions are intended to strengthen lower bounds for $W_1$4 or $W_1$5. In the 2019 two-sided linkage paper, Theorem 2 produces an $W_1$6 CDC, and the paper states improvements for

$W_1$7

For $W_1$8, the stated lower bounds are

$W_1$9

$$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q_1\}, \qquad W_2=\{\operatorname{Im}(Q\mid U): Q\in \mathcal Q_2,\ U\in \mathcal V\},$$0

$$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q_1\}, \qquad W_2=\{\operatorname{Im}(Q\mid U): Q\in \mathcal Q_2,\ U\in \mathcal V\},$$1

(He, 2019).

The several-parallel-lifted-MRD paper gives a multi-parallel lower bound for $$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q_1\}, \qquad W_2=\{\operatorname{Im}(Q\mid U): Q\in \mathcal Q_2,\ U\in \mathcal V\},$$2 and reports, for example, that when $$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q_1\}, \qquad W_2=\{\operatorname{Im}(Q\mid U): Q\in \mathcal Q_2,\ U\in \mathcal V\},$$3, $$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q_1\}, \qquad W_2=\{\operatorname{Im}(Q\mid U): Q\in \mathcal Q_2,\ U\in \mathcal V\},$$4, $$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q_1\}, \qquad W_2=\{\operatorname{Im}(Q\mid U): Q\in \mathcal Q_2,\ U\in \mathcal V\},$$5,

$$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q_1\}, \qquad W_2=\{\operatorname{Im}(Q\mid U): Q\in \mathcal Q_2,\ U\in \mathcal V\},$$6

For $$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q_1\}, \qquad W_2=\{\operatorname{Im}(Q\mid U): Q\in \mathcal Q_2,\ U\in \mathcal V\},$$7, using

$$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q_1\}, \qquad W_2=\{\operatorname{Im}(Q\mid U): Q\in \mathcal Q_2,\ U\in \mathcal V\},$$8

the paper obtains

$$W_1=\{\operatorname{Im}(U\mid Q): U\in \mathcal U,\ Q\in \mathcal Q_1\}, \qquad W_2=\{\operatorname{Im}(Q\mid U): Q\in \mathcal Q_2,\ U\in \mathcal V\},$$9

larger than

$\mathcal G_q(n,k)$00

(He et al., 2019).

The generalized linkage paper highlights the benchmark $\mathcal G_q(n,k)$01. For $\mathcal G_q(n,k)$02, it reports an improvement from $\mathcal G_q(n,k)$03 to $\mathcal G_q(n,k)$04, while the parallel linkage construction gave only $\mathcal G_q(n,k)$05 (Heinlein, 2019). The parallel multilevel paper states that the ratio between the new lower bound and the known upper bound for $\mathcal G_q(n,k)$06-CDCs is greater than $\mathcal G_q(n,k)$07 for any prime power $\mathcal G_q(n,k)$08 and any $\mathcal G_q(n,k)$09 (Liu et al., 2019).

Within the specific parallel mixed dimension framework, the bilateral refinement gives explicit new values. In Example 6, with

$\mathcal G_q(n,k)$10

the paper uses an $\mathcal G_q(n,k)$11 MDDC with

$\mathcal G_q(n,k)$12

and obtains

$\mathcal G_q(n,k)$13

The new bilateral part contributes

$\mathcal G_q(n,k)$14

so that

$\mathcal G_q(n,k)$15

improving the previous lower bound $\mathcal G_q(n,k)$16 (Li et al., 10 Jul 2025).

In Example 7, for

$\mathcal G_q(n,k)$17

the paper reports

$\mathcal G_q(n,k)$18

hence

$\mathcal G_q(n,k)$19

improving $\mathcal G_q(n,k)$20 (Li et al., 10 Jul 2025). The same paper states that Table 1 gives at least 49 new lower bounds (Li et al., 10 Jul 2025).

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 (Li et al., 10 Jul 2025). 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 (He, 2019). 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 (Heinlein, 2019).

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 (Li et al., 10 Jul 2025). A plausible implication is that the construction is best understood not as a departure from the CDC framework but as an overview of three earlier lines: two-sided linkage, several parallel lifted MRD constructions, and multilevel/Ferrers-based refinement (He, 2019, He et al., 2019, Liu et al., 2019).

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

$\mathcal G_q(n,k)$21

and that for $\mathcal G_q(n,k)$22 it uses rank-restricted rank-metric codes instead (He, 2019). The generalized linkage framework introduces the parameter $\mathcal G_q(n,k)$23 to interpolate between improved linkage and parallel linkage, and its flexibility can strictly improve both older constructions for infinite parameter families (Heinlein, 2019). The bilateral refinement then densifies the parallel mixed dimension skeleton by inserting additional codewords $\mathcal G_q(n,k)$24 while keeping the distance conditions intact (Li et al., 10 Jul 2025).

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 (Li et al., 10 Jul 2025).