Equivalence Classes in Pre-Transformed Polar Codes

• The paper introduces an equivalence-class framework that preserves Hamming weight distributions across pre-transformed polar codes.
• It employs cyclic shifts of generator rows to partition pre-transformation matrices, significantly reducing the search complexity for weight analysis.
• Empirical results show that optimizing the expanded information set can yield up to a 16-fold speed-up in weight distribution computations for PAC 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 (Liu et al., 12 Jan 2026).

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 (Moradi, 2023).

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 $u$ (Liu et al., 12 Jan 2026). 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 (Liu et al., 12 Jan 2026).

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 (Liu et al., 12 Jan 2026).

A canonical procedure for this optimization is as follows (see (Liu et al., 12 Jan 2026)):

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 (Liu et al., 12 Jan 2026).

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 (Moradi, 2023), 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 (Lulei et al., 24 Apr 2025).

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.