GBMC: Generalized Bilateral Multilevel Construction

• GBMC is a coding-theoretic framework that combines outer binary constant-weight codes with inner generalized bilateral Ferrers-diagram rank-metric codes to construct large constant dimension codes.
• It integrates dual pivot structures and a specialized rank-restricted upper-right block to ensure robust subspace distances and improved lower bounds in network coding.
• GBMC augments the parallel mixed dimension construction by adding a third distance-compatible layer, further enhancing performance and code size.

Searching arXiv for the primary paper and closely related constructions. Searching for recent arXiv records on generalized bilateral multilevel construction, bilateral multilevel construction, and parallel mixed dimension construction. Generalized Bilateral Multilevel Construction (GBMC) is a coding-theoretic framework for constructing constant dimension codes in the Grassmannian by combining an outer binary constant-weight code on bilateral identifying vectors with inner generalized bilateral Ferrers-diagram rank-metric codes. Its purpose is to produce large $(n,2\delta,\{k\})_q$ constant dimension codes and hence improved lower bounds on $A_q(n,2\delta,\{k\})$. In the 2025 formulation, GBMC is further used to enhance the parallel mixed dimension construction by adding a third, distance-compatible layer of codewords (Li et al., 10 Jul 2025).

1. Problem setting in subspace coding

Let $q$ be a prime power and $\mathbb{F}_q^n$ the $n$-dimensional vector space over $\mathbb{F}_q$. The set of all subspaces is $\mathcal{P}_q(n)$, and the Grassmannian $\mathcal{G}_q(n,k)$ is the set of all $k$-dimensional subspaces. Its cardinality is the Gaussian binomial coefficient

$\left[\begin{matrix}n\k\end{matrix}\right]_q = \prod_{i=0}^{k-1}\frac{q^{n-i}-1}{q^{k-i}-1}.$

For subspaces $A_q(n,2\delta,\{k\})$0, the subspace distance is

$A_q(n,2\delta,\{k\})$1

On $A_q(n,2\delta,\{k\})$2, this simplifies to

$A_q(n,2\delta,\{k\})$3

An $A_q(n,2\delta,\{k\})$4 constant dimension code (CDC) is a subset $A_q(n,2\delta,\{k\})$5 such that

$A_q(n,2\delta,\{k\})$6

The central extremal quantity is

$A_q(n,2\delta,\{k\})$7

GBMC addresses the standard lower-bound problem for $A_q(n,2\delta,\{k\})$8. This problem is important in random linear network coding, where codewords are subspaces and subspace distance measures the effect of dimension errors and erasures.

2. Bilateral identifying vectors

The construction builds on three pivot-pattern notions associated with subspaces in echelon form.

Notion	Source form	Distance role
Identifying vector $A_q(n,2\delta,\{k\})$9	Reduced row echelon form $q$0	$q$1
Inverse identifying vector $q$2	Reduced row inverse echelon form $q$3	$q$4
Bilateral identifying vector $q$5	Left identifying block, zero middle block, right inverse identifying block	Outer code of GBMC

If two subspaces have the same identifying vector, then deleting the pivot columns from their reduced row echelon forms converts subspace distance into rank distance: $q$6 The analogous statement holds for equal inverse identifying vectors. GBMC combines these two viewpoints into a single pattern class.

A bilateral identifying vector $q$7 of length $q$8 is defined via a partition

$q$9

into three blocks

$\mathbb{F}_q^n$0

where $\mathbb{F}_q^n$1 is an identifying vector, $\mathbb{F}_q^n$2 is an inverse identifying vector, and $\mathbb{F}_q^n$3 is the all-zero vector. If

$\mathbb{F}_q^n$4

then the ordered triple

$\mathbb{F}_q^n$5

is called the type of $\mathbb{F}_q^n$6.

This bilateral structure separates a left RREF-controlled block and a right inverse-RREF-controlled block, while the middle block carries no pivots. In the version compatible with the parallel mixed dimension construction, the paper imposes the additional range condition

$\mathbb{F}_q^n$7

which is later used to maintain distance from the parallel mixed-dimension component (Li et al., 10 Jul 2025).

3. Generalized bilateral Ferrers geometry

Given a bilateral identifying vector $\mathbb{F}_q^n$8 of type $\mathbb{F}_q^n$9, the generalized bilateral echelon Ferrers form $n$0 is a $n$1 matrix with five structural blocks.

The upper-left $n$2 block is the usual echelon Ferrers form $n$3. The lower-right $n$4 block is the inverse echelon Ferrers form $n$5. The lower-left $n$6 block is zero. In the upper-right $n$7 block, the columns that are pivot columns of $n$8 are filled with zeros, while the remaining columns are free positions. The middle block, corresponding to $n$9, is a full $\mathbb{F}_q$0 Ferrers diagram.

The set of dot positions in $\mathbb{F}_q$1 is the generalized bilateral Ferrers diagram $\mathbb{F}_q$2. It decomposes as

$\mathbb{F}_q$3

where $\mathbb{F}_q$4 and $\mathbb{F}_q$5 arise from the left and right pivot structures, $\mathbb{F}_q$6 is an $\mathbb{F}_q$7 full Ferrers diagram, $\mathbb{F}_q$8 is an $\mathbb{F}_q$9 full diagram, and $\mathcal{P}_q(n)$0 is an $\mathcal{P}_q(n)$1 full diagram.

A distinguished submatrix of a matrix $\mathcal{P}_q(n)$2 supported on $\mathcal{P}_q(n)$3 is

$\mathcal{P}_q(n)$4

the $\mathcal{P}_q(n)$5 upper-right block associated with $\mathcal{P}_q(n)$6. This block is central in the compatibility conditions with the parallel mixed dimension construction, because its rank is explicitly constrained in Theorems 4–7.

An $\mathcal{P}_q(n)$7 generalized bilateral Ferrers diagram rank-metric code (GB-FD code) is a set of matrices over $\mathcal{P}_q(n)$8 having shape $\mathcal{P}_q(n)$9 and minimum rank distance $\mathcal{G}_q(n,k)$0. Proposition 1 gives a lower bound for such codes under a rank restriction on a chosen upper-right submatrix. Structurally, the lower bound combines two MRD-type contributions from full Ferrers blocks and a term $\mathcal{G}_q(n,k)$1 controlling the rank-restricted block. This is the technical device that makes the later augmentation of the parallel mixed dimension construction possible.

4. The multilevel mechanism of GBMC

The outer level of GBMC is a set $\mathcal{G}_q(n,k)$2 of bilateral identifying vectors. The inner level, for each $\mathcal{G}_q(n,k)$3, is a GB-FD code $\mathcal{G}_q(n,k)$4 supported on $\mathcal{G}_q(n,k)$5. The lifting operation $\mathcal{G}_q(n,k)$6 inserts the matrices of $\mathcal{G}_q(n,k)$7 into the generalized bilateral echelon Ferrers form $\mathcal{G}_q(n,k)$8 and takes their rowspaces.

Two distance principles justify the construction. If $\mathcal{G}_q(n,k)$9 and $k$0 are bilateral identifying vectors of the same type and $k$1 are the corresponding rowspaces, then

$k$2

If $k$3 and $k$4 are based on the same bilateral identifying vector $k$5, and $k$6 are obtained by deleting the pivot columns from the left and right pivot blocks, then

$k$7

Construction 4 may therefore be stated as follows. Let $k$8 be a set of bilateral identifying vectors forming a binary constant-weight code of length $k$9, weight $\left[\begin{matrix}n\k\end{matrix}\right]_q = \prod_{i=0}^{k-1}\frac{q^{n-i}-1}{q^{k-i}-1}.$0, and minimum Hamming distance $\left[\begin{matrix}n\k\end{matrix}\right]_q = \prod_{i=0}^{k-1}\frac{q^{n-i}-1}{q^{k-i}-1}.$1, and assume all vectors in $\left[\begin{matrix}n\k\end{matrix}\right]_q = \prod_{i=0}^{k-1}\frac{q^{n-i}-1}{q^{k-i}-1}.$2 have the same type. For each $\left[\begin{matrix}n\k\end{matrix}\right]_q = \prod_{i=0}^{k-1}\frac{q^{n-i}-1}{q^{k-i}-1}.$3, suppose there exists an $\left[\begin{matrix}n\k\end{matrix}\right]_q = \prod_{i=0}^{k-1}\frac{q^{n-i}-1}{q^{k-i}-1}.$4 GB-FD code $\left[\begin{matrix}n\k\end{matrix}\right]_q = \prod_{i=0}^{k-1}\frac{q^{n-i}-1}{q^{k-i}-1}.$5. Then each lifted family $\left[\begin{matrix}n\k\end{matrix}\right]_q = \prod_{i=0}^{k-1}\frac{q^{n-i}-1}{q^{k-i}-1}.$6 is an $\left[\begin{matrix}n\k\end{matrix}\right]_q = \prod_{i=0}^{k-1}\frac{q^{n-i}-1}{q^{k-i}-1}.$7 CDC, and the union

$\left[\begin{matrix}n\k\end{matrix}\right]_q = \prod_{i=0}^{k-1}\frac{q^{n-i}-1}{q^{k-i}-1}.$8

is an $\left[\begin{matrix}n\k\end{matrix}\right]_q = \prod_{i=0}^{k-1}\frac{q^{n-i}-1}{q^{k-i}-1}.$9 CDC (Li et al., 10 Jul 2025).

The multilevel character is exact: the outer code controls distance through Hamming separation of bilateral patterns, and the inner code controls distance within each pattern class through rank distance. This is the bilateral analogue of the classical multilevel paradigm based on identifying vectors and Ferrers-diagram rank-metric codes, but with simultaneous left and right pivot structure.

5. Coupling with the parallel mixed dimension construction

The 2025 development of GBMC is not limited to a standalone union over bilateral patterns. It is integrated with the parallel mixed dimension construction by adjoining a third component $A_q(n,2\delta,\{k\})$00 to the existing two-part code $A_q(n,2\delta,\{k\})$01. The aim is to enlarge the code while preserving minimum subspace distance $A_q(n,2\delta,\{k\})$02.

In the general version, the parameters are

$A_q(n,2\delta,\{k\})$03

One chooses a set $A_q(n,2\delta,\{k\})$04 of bilateral identifying vectors of weight $A_q(n,2\delta,\{k\})$05, minimum Hamming distance $A_q(n,2\delta,\{k\})$06, and common type $A_q(n,2\delta,\{k\})$07, satisfying

$A_q(n,2\delta,\{k\})$08

for every $A_q(n,2\delta,\{k\})$09. For each $A_q(n,2\delta,\{k\})$10, one also requires an $A_q(n,2\delta,\{k\})$11 GB-FD code $A_q(n,2\delta,\{k\})$12 such that

$A_q(n,2\delta,\{k\})$13

Under these conditions,

$A_q(n,2\delta,\{k\})$14

is an $A_q(n,2\delta,\{k\})$15 CDC, and

$A_q(n,2\delta,\{k\})$16

is an $A_q(n,2\delta,\{k\})$17 CDC.

The proof separates three interactions. Distances inside $A_q(n,2\delta,\{k\})$18 are handled by the GBMC distance lemmas. Distances between $A_q(n,2\delta,\{k\})$19 and $A_q(n,2\delta,\{k\})$20 use identifying-vector arguments and the weight condition $A_q(n,2\delta,\{k\})$21. Distances between $A_q(n,2\delta,\{k\})$22 and $A_q(n,2\delta,\{k\})$23 use inverse identifying vectors together with the rank restriction on $A_q(n,2\delta,\{k\})$24.

A simplified version replaces $A_q(n,2\delta,\{k\})$25 by $A_q(n,2\delta,\{k\})$26: $A_q(n,2\delta,\{k\})$27 The same strategy applies, again producing a larger CDC by adjoining a GBMC layer to the parallel mixed-dimension construction (Li et al., 10 Jul 2025).

6. Explicit bounds and representative improvements

Theorems 6 and 7 make the construction explicit by choosing grid families of bilateral identifying vectors

$A_q(n,2\delta,\{k\})$28

whose left and right blocks contain consecutive ones in sliding positions. The parameters $A_q(n,2\delta,\{k\})$29 and $A_q(n,2\delta,\{k\})$30 range over rectangles determined by $A_q(n,2\delta,\{k\})$31 and $A_q(n,2\delta,\{k\})$32, and the weights $A_q(n,2\delta,\{k\})$33 are chosen so that the Hamming distance between distinct $A_q(n,2\delta,\{k\})$34 is at least $A_q(n,2\delta,\{k\})$35. For each such vector, Proposition 1 supplies a compatible GB-FD code, and summing over the grid yields the size of the added component $A_q(n,2\delta,\{k\})$36.

The resulting lower bounds on $A_q(n,2\delta,\{k\})$37 have a characteristic form: the original parallel mixed-dimension contribution

$A_q(n,2\delta,\{k\})$38

is augmented by a GBMC term obtained by summing the sizes of the lifted GB-FD codes over all bilateral patterns in the chosen grid.

Two examples illustrate the gain:

Parameters	Previous value from the parallel mixed dimension part	Improved value after adding GBMC
$A_q(n,2\delta,\{k\})$39	$A_q(n,2\delta,\{k\})$40	$A_q(n,2\delta,\{k\})$41
$A_q(n,2\delta,\{k\})$42	$A_q(n,2\delta,\{k\})$43	$A_q(n,2\delta,\{k\})$44

In the first case, the added GBMC layer contributes $A_q(n,2\delta,\{k\})$45. In the second, it contributes $A_q(n,2\delta,\{k\})$46. The paper further states that at least 49 parameter sets $A_q(n,2\delta,\{k\})$47 are improved, and it derives infinite families of improved lower bounds through corollaries specializing the general theorems (Li et al., 10 Jul 2025).

These numerical results clarify the operational meaning of the generalized construction. GBMC is not merely a reformulation of existing multilevel methods; it acts as a combinatorially controlled augmentation layer whose contribution is additive at the code-size level and distance-compatible at the Grassmannian level.

7. Conceptual position and common interpretive points

GBMC belongs to a sequence of subspace-code constructions that progressively enlarge the admissible pivot-pattern geometry. The Etzion–Silberstein multilevel construction uses identifying vectors and Ferrers-diagram rank-metric codes. The inverse multilevel construction replaces identifying vectors by inverse identifying vectors. The double multilevel construction combines the two. The bilateral multilevel construction introduces bilateral identifying vectors, and the generalized bilateral multilevel construction extends this by allowing generalized bilateral Ferrers diagrams and a rank-restricted upper-right block. The 2025 synthesis then merges GBMC with the mixed dimension and parallel mixed dimension constructions.

Several interpretive points are essential.

First, a bilateral identifying vector is not an arbitrary concatenation of binary blocks. Its middle block is all-zero, its left and right blocks have different echelon-theoretic roles, and in Construction 4 all outer vectors must have the same type.

Second, compatibility with the parallel mixed dimension construction is not automatic. It requires the weight window

$A_q(n,2\delta,\{k\})$48

and, in the integrated theorems, the rank restriction

$A_q(n,2\delta,\{k\})$49

Without these conditions, the additional GBMC layer need not maintain distance from the pre-existing parallel components.

Third, the generalized construction is broader than a simple left-right concatenation of classical Ferrers-diagram pieces. The blocks $A_q(n,2\delta,\{k\})$50, $A_q(n,2\delta,\{k\})$51, and $A_q(n,2\delta,\{k\})$52 create a mixed geometry in which full rectangular regions interact with left and right pivot structures. This is precisely what enables Proposition 1 and the later coupling with mixed-dimension outer codes.

A broader coding-theoretic perspective suggests that GBMC is part of a general multilevel design pattern in which an outer discrete pattern code is coupled to inner algebraic constituents. Multilevel lattice constructions from nested binary codes exhibit the same separation of roles across levels (Kositwattanarerk et al., 2013), and generalized Construction D' for LDPC lattices similarly enlarges the admissible design space by relaxing structural constraints while preserving multilevel behavior (Silva et al., 2017). This suggests a methodological affinity rather than a direct equivalence: in each case, multilevel organization is used to increase constructive flexibility without surrendering a rigorous distance or decoding analysis.

Within constant-dimension coding, GBMC is therefore best understood as a highly structured enlargement of the multilevel paradigm: it replaces single-sided pivot patterns by bilateral ones, replaces ordinary Ferrers supports by generalized bilateral supports, and, in its most recent form, turns those ingredients into an efficient augmentation of the parallel mixed dimension construction.