Binary linear block codes (BLBCs) are essential to modern communication, but their diverse structures often require tailor-made decoders, increasing complexity. This work introduces enhanced polar decoding ( $\textsf {PD}^{+}$ ), a universal soft decoding algorithm that transforms any BLBC into a polar-like code compatible with efficient polar code decoders such as successive cancellation list (SCL) decoding. Key innovations in $\textsf {PD}^{+}$ include pruning polar kernels, shortening codes, and leveraging a simulated annealing algorithm to optimize transformations. These enable $\textsf {PD}^{+}$ to achieve competitive or superior performance to state-of-the-art algorithms like OSD and GRAND across various codes, including extended BCH, extended Golay, and binary quadratic residue codes, with significantly lower complexity. Moreover, $\textsf {PD}^{+}$ is designed to be forward-compatible with advancements in polar code decoding techniques and AI-driven search methods, making it a robust and versatile solution for universal BLBC decoding in both present and future systems.Binary linear block codes (BLBCs) are essential to modern communication, but their diverse structures often require tailor-made decoders, increasing complexity. This work introduces enhanced polar decoding ($\textsf {PD}^{+}$), a universal soft decoding algorithm that transforms any BLBC into a polar-like code compatible with efficient polar code decoders such as successive cancellation list (SCL) decoding. Key innovations in$\textsf {PD}^{+}$include pruning polar kernels, shortening codes, and leveraging a simulated annealing algorithm to optimize transformations. These enable$\textsf {PD}^{+}$to achieve competitive or superior performance to state-of-the-art algorithms like OSD and GRAND across various codes, including extended BCH, extended Golay, and binary quadratic residue codes, with significantly lower complexity. Moreover,$\textsf {PD}^{+}$is designed to be forward-compatible with advancements in polar code decoding techniques and AI-driven search methods, making it a robust and versatile solution for universal BLBC decoding in both present and future systems.

arXiv:2501.07279IEEE Transactions on CommunicationsJul 8, 20255 citations

The Correcting Capacities of Linear Codes

M. V. Kozlov

Soviet physics. DokladyNov 1, 19693 citations

A decoding algorithm for linear codes

Martin Bossert, F. Hergert

Lecture notes in computer scienceJan 1, 19862 citations

On Hull-Variation Problem of Equivalent Linear Codes

Hao Chen

The intersection ${\bf C}\bigcap {\bf C}^{\perp}$ ( ${\bf C}\bigcap {\bf C}^{\perp_h}$ ) of a linear code ${\bf C}$ and its Euclidean dual ${\bf C}^{\perp}$ (Hermitian dual ${\bf C}^{\perp_h}$ ) is called the Euclidean (Hermitian) hull of this code. It is natural to consider the hull-variation problem when a linear code ${\bf C}$ is transformed to an equivalent code ${\bf v} \cdot {\bf C}$ . In this paper we introduce the maximal hull dimension as an invariant of a linear code with respect to the equivalent transformations. Then some basic properties of the maximal hull dimension are studied. We prove that for a nonnegative integer $h$ satisfying $0 \leq h \leq n-1$ , a linear $[2n, n]_q$ self-dual code is equivalent to a linear $h$ -dimension hull code. On the opposite direction we prove that a linear LCD code over ${\bf F}_{2^s}$ satisfying $d\geq 2$ and $d^{\perp} \geq 2$ is equivalent to a linear one-dimension hull code under a weak condition. Several new families of LCD negacyclic codes and LCD BCH codes over ${\bf F}_3$ are also constructed. Our method can be applied to the generalized Reed-Solomon codes and the generalized twisted Reed-Solomon codes to construct arbitrary dimension hull MDS codes. Some new entanglement-assisted quantum error-correction (EAQEC) codes including MDS and almost MDS EAQEC codes are constructed. Many EAQEC codes over small fields are constructed from optimal Hermitian self-dual codes.The intersection ${\bf C}\bigcap {\bf C}^{\perp}$ (${\bf C}\bigcap {\bf C}^{\perp_h}$) of a linear code ${\bf C}$ and its Euclidean dual ${\bf C}^{\perp}$ (Hermitian dual ${\bf C}^{\perp_h}$) is called the Euclidean (Hermitian) hull of this code. It is natural to consider the hull-variation problem when a linear code ${\bf C}$ is transformed to an equivalent code ${\bf v} \cdot {\bf C}$. In this paper we introduce the maximal hull dimension as an invariant of a linear code with respect to the equivalent transformations. Then some basic properties of the maximal hull dimension are studied. We prove that for a nonnegative integer $h$ satisfying $0 \leq h \leq n-1$, a linear $[2n, n]_q$ self-dual code is equivalent to a linear $h$-dimension hull code. On the opposite direction we prove that a linear LCD code over ${\bf F}_{2^s}$ satisfying $d\geq 2$ and $d^{\perp} \geq 2$ is equivalent to a linear one-dimension hull code under a weak condition. Several new families of LCD negacyclic codes and LCD BCH codes over ${\bf F}_3$ are also constructed. Our method can be applied to the generalized Reed-Solomon codes and the generalized twisted Reed-Solomon codes to construct arbitrary dimension hull MDS codes. Some new entanglement-assisted quantum error-correction (EAQEC) codes including MDS and almost MDS EAQEC codes are constructed. Many EAQEC codes over small fields are constructed from optimal Hermitian self-dual codes.

arXiv:2206.14516Jun 29, 20220 citations

5G LDPC Linear Transformer for Channel Decoding

Hernandez, Mario, Fernando Piñero

This work introduces a novel, fully differentiable linear-time complexity transformer decoder and a transformer decoder to correct 5G New Radio (NR) LDPC. We propose a scalable approach to decode linear block codes with $O(n)$ complexity rather than $O(n^2)$ for regular transformers. The architectures' performances are compared to Belief Propagation (BP), the production-level decoding algorithm used for 5G New Radio (NR) LDPC codes. We achieve bit error rate performance that matches a regular Transformer decoder and surpases one iteration BP, also achieving competitive time performance against BP, even for larger block codes. We utilize Sionna, Nvidia's 5G & 6G physical layer research software, for reproducible results.This work introduces a novel, fully differentiable linear-time complexity transformer decoder and a transformer decoder to correct 5G New Radio (NR) LDPC. We propose a scalable approach to decode linear block codes with $O(n)$ complexity rather than $O(n^2)$ for regular transformers. The architectures' performances are compared to Belief Propagation (BP), the production-level decoding algorithm used for 5G New Radio (NR) LDPC codes. We achieve bit error rate performance that matches a regular Transformer decoder and surpases one iteration BP, also achieving competitive time performance against BP, even for larger block codes. We utilize Sionna, Nvidia's 5G & 6G physical layer research software, for reproducible results.

arXiv:2501.14102Jan 23, 20250 citations

Decoding Methods for Linear Codes

Kenny Barrese

Jun 4, 20080 citations

\textsf {PD}^{+}

), a universal soft decoding algorithm that transforms any BLBC into a polar-like code compatible with efficient polar code decoders such as successive cancellation list (SCL) decoding. Key innovations in

\textsf {PD}^{+}

include pruning polar kernels, shortening codes, and leveraging a simulated annealing algorithm to optimize transformations. These enable

\textsf {PD}^{+}

to achieve competitive or superior performance to state-of-the-art algorithms like OSD and GRAND across various codes, including extended BCH, extended Golay, and binary quadratic residue codes, with significantly lower complexity. Moreover,

\textsf {PD}^{+}

is designed to be forward-compatible with advancements in polar code decoding techniques and AI-driven search methods, making it a robust and versatile solution for universal BLBC decoding in both present and future systems.

The intersection

{\bf C}\bigcap {\bf C}^{\perp}

(

{\bf C}\bigcap {\bf C}^{\perp_h}

) of a linear code

{\bf C}

and its Euclidean dual

{\bf C}^{\perp}

(Hermitian dual

{\bf C}^{\perp_h}

) is called the Euclidean (Hermitian) hull of this code. It is natural to consider the hull-variation problem when a linear code

{\bf C}

is transformed to an equivalent code

{\bf v} \cdot {\bf C}

. In this paper we introduce the maximal hull dimension as an invariant of a linear code with respect to the equivalent transformations. Then some basic properties of the maximal hull dimension are studied. We prove that for a nonnegative integer

h

satisfying

0 \leq h \leq n-1

, a linear

[2n, n]_q

self-dual code is equivalent to a linear

h

-dimension hull code. On the opposite direction we prove that a linear LCD code over

{\bf F}_{2^s}

satisfying

d\geq 2

and

d^{\perp} \geq 2

is equivalent to a linear one-dimension hull code under a weak condition. Several new families of LCD negacyclic codes and LCD BCH codes over

{\bf F}_3

are also constructed. Our method can be applied to the generalized Reed-Solomon codes and the generalized twisted Reed-Solomon codes to construct arbitrary dimension hull MDS codes. Some new entanglement-assisted quantum error-correction (EAQEC) codes including MDS and almost MDS EAQEC codes are constructed. Many EAQEC codes over small fields are constructed from optimal Hermitian self-dual codes.

O(n)

complexity rather than

O(n^2)

for regular transformers. The architectures' performances are compared to Belief Propagation (BP), the production-level decoding algorithm used for 5G New Radio (NR) LDPC codes. We achieve bit error rate performance that matches a regular Transformer decoder and surpases one iteration BP, also achieving competitive time performance against BP, even for larger block codes. We utilize Sionna, Nvidia's 5G & 6G physical layer research software, for reproducible results.