---
title: Equivalence Classes in Pre-Transformed Polar Codes
url: https://www.emergentmind.com/topics/equivalence-classes-for-pre-transformed-polar-codes
type: topic
---

# Equivalence Classes in Pre-Transformed Polar Codes

Pre-transformed polar codes are an important class of linear block codes that generalize the classical polar construction by applying an invertible, typically upper-triangular, pre-transformation matrix before the standard polar encoding. The introduction of equivalence classes for pre-transformation matrices provides a structural basis to optimize algorithmic complexity, particularly for calculating code weight distributions, without altering core code properties like the minimum distance. These equivalence classes capture the inherent symmetries of pre-transformed polar codes under transformations generated by cyclic shifts of rows of the polar kernel, leading to profound practical simplifications in code analysis and implementation.

## 1. Pre-Transformations and the Polar Code Structure

Let $N=2^n$ denote the code length, and let $G_n = F^{\otimes n}$ be the standard polar generator matrix, with $F = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$. A pre-transformation is defined by an $N\times N$ upper-triangular binary matrix $T$ with ones on the diagonal. The generator matrix for pre-transformed polar codes is then given by $G = T \cdot G_n$, and codewords are generated as $c = u_{\,\mathcal{I}} \, T \, G_n$ with $u_{\mathcal{F}} = 0$, where $\mathcal{I}$ and $\mathcal{F}$ are the information and frozen sets, respectively [2601.07515].

This structure generalizes polarization-adjusted convolutional (PAC) codes, where the pre-transformation $T$ is a Toeplitz matrix determined by a binary connection polynomial $g(D)$. Every PAC code is thus a special case of pre-transformed polar codes and may be viewed as a serial concatenation of a cyclic inner code and an outer polar- or Reed–Muller-like code [2304.01463].

## 2. Equivalence Relation on Pre-Transformation Matrices

To exploit symmetries in code structure and computational complexity, an equivalence relation on pre-transformation matrices is defined. Two pre-transform matrices $T$ and $T'$ are declared equivalent, $T \sim T'$, if there exists an $i \in \{1, \ldots, N\}$ such that $T' = T \cdot T(g_i)$, where $T(g_i)$ is the shift-generated pre-transform based on the $i$-th row $g_i$ of $G_n$:

\[
T(g_i)_{j,*} = o_{j-1}(g_i)
\]

Here, $o_{s}(u)$ denotes the cyclic shift by $s$ positions of the vector $u$ [2601.07515]. This equivalence relation is reflexive, symmetric, and transitive, partitioning the space of pre-transforms into classes of size at most $N$.

## 3. Invariance of Weight Distribution within Equivalence Classes

A central theoretical result is that codes generated by any two equivalent pre-transformation matrices share the identical Hamming weight distribution. That is, for all $w$,

\[
W_{C(T)}(w) = W_{C(T')}(w)
\]
whenever $T' = T \cdot T(g_i)$

This follows because right multiplication by $T(g_i)$ cyclically shifts the generator rows, so each codeword of $C(T)$ maps bijectively to a codeword of $C(T')$ by a cyclic shift, which preserves Hamming weight. Thus, properties such as minimum distance, weight spectrum, and error rate bounds remain invariant under this group action [2601.07515].

## 4. Optimization of the Expanded Information Set and Algorithmic Implications

Despite equal weight distributions across an equivalence class, the recursive complexity of algorithms—such as the Parity-Consistent Decomposition (PCD) method for weight distribution computation—depends on the size of the so-called "Expanded Information Set" $P(T)$. For a given pre-transformation $T$, the cost of computing the weight distribution via PCD is $O(2^{A(T)}\,\mathrm{poly}(N))$ with $A(T) = |P(T)|$.

By exhaustively searching only over the at most $N$ members of an equivalence class, one finds the $T^*$ that minimizes $A(T)$. This minimizes the branching factor and thus exponential complexity, yielding speed-ups proportional to $2^{\Delta A}$, where $\Delta A$ is the reduction in the expanded information set size compared to an arbitrary class representative [2601.07515].

A canonical procedure for this optimization is as follows (see [2601.07515]):

| Step | Operation                                              | Complexity         |
|------|--------------------------------------------------------|--------------------|
| 1    | Precompute all $g_i$ (rows of $G_n$)                   | $O(N^2)$           |
| 2    | For each $i=1, \ldots, N$:                             |                    |
| 2a   | --- Form $T_i = T_0 \cdot T(g_i)$                      | $O(N^2)$           |
| 2b   | --- Compute $P_i$, $A_i = |P_i|$ using Algorithm 1      | $O(N^2)$           |
| 3    | Choose $T^*$ with minimal $A_i$                        | $O(N)$             |
| 4    | Compute WD by PCD with $T^*$                           | $O(2^{A_*}\mathrm{poly}(N))$ |

This reduces the naïve search over $2^{N(N-1)/2}$ upper-triangular $T$ to evaluating only $N$ candidates.

## 5. Examples and Empirical Results

Empirical results demonstrate the significance of the equivalence-class method. In a length-128 PAC code, application of this method reduced the expanded information set by 4 bits, resulting in a 16-fold (i.e., $2^4$) speed-up in PCD-based weight distribution calculation. Over random instances at lengths 128 and 256, the probability of any reduction in $A$ increased from approximately $6.7\%$ (without equivalence optimization) to $21\%$ using the equivalence-class strategy [2601.07515].

Explicit examples with small $N$ (e.g., $N=8$) confirm that, for $T_0$ and $T_0 T(g_4)$, the complete weight distribution matches, and codewords relate through cyclic shifts.

## 6. Broader Consequences for Polar Coding Theory

The equivalence-class structure reinforces and generalizes classical results about code automorphisms and isomorphisms within coding theory. It rigorously underpins the observed performance equivalence of cyclically shifted pre-transform variants and justifies their use in practical code design and algorithm optimization. For PAC codes, this theory confirms their algebraic equivalence to concatenated cyclic-polar schemas [2304.01463], explaining their improved minimum distance and empirical error rates over conventional polar- and RM-like designs. In emerging decoding approaches such as Subcode Ensemble Decoding (ScED), the pre-transform framework enables the construction of ensembles and subcode partitions without altering critical code metrics, permitting fine-grained trade-offs in complexity and performance [2504.17511].

A plausible implication is that further advances in polar code optimization—whether for error analysis, implementation efficiency, or list-decoding performance—will systematically invoke this equivalence-class framework as a foundation for both theoretical results and engineering heuristics.

Source: https://www.emergentmind.com/topics/equivalence-classes-for-pre-transformed-polar-codes