We present several new provable algorithms for two variants of the code equivalence problem on linear error-correcting codes, the Linear Code Equivalence Problem (LCE) and the Permutation Code Equivalence Problem (PCE). Specifically, for arbitrary codes of block length $n$ and dimension $k$ over any finite field $\mathbb{F}_{q}$ , we show: 1)A deterministic algorithm running in $2^{n+o(n+q)}$ time for LCE. 2)A randomized algorithm running in $2^{n / 2+o(n+q)}$ time for LCE and PCE. 3)A quantum algorithm running in $2^{n / 3+o(n+q)}$ time for LCE and PCE. The second two algorithms improve on recent work of Nowakowski (PQCrypto 2025), which gave algorithms with similar running times but only for code equivalence on random codes and only over fields of order $q \geq 7$ .We present several new provable algorithms for two variants of the code equivalence problem on linear error-correcting codes, the Linear Code Equivalence Problem (LCE) and the Permutation Code Equivalence Problem (PCE). Specifically, for arbitrary codes of block length$n$and dimension$k$over any finite field$\mathbb{F}_{q}$, we show: 1)A deterministic algorithm running in$2^{n+o(n+q)}$time for LCE. 2)A randomized algorithm running in$2^{n / 2+o(n+q)}$time for LCE and PCE. 3)A quantum algorithm running in$2^{n / 3+o(n+q)}$time for LCE and PCE. The second two algorithms improve on recent work of Nowakowski (PQCrypto 2025), which gave algorithms with similar running times but only for code equivalence on random codes and only over fields of order$q \geq 7$.

IEEE International Symposium on Information Theory (ISIT)Jun 22, 20251 citation

Linear-Time Encodable and Decodable Quantum Error-Correcting Codes

Adam Wills, Ting-Chun Lin, Rachel Yun Zhang +1 more

Recent years have seen rapid development in the subject of quantum coding theory, with breakthroughs on many exciting classes of codes, including quantum LDPC codes, quantum locally testable codes, and quantum codes with interesting transversal gates. However, a natural class of quantum codes, which has been well-studied classically, has not yet been treated: those which can be quickly encoded and decoded. This problem concerns the channel capacity setting, where a noise channel sits between perfect encoding and unencoding/decoding operations; this is the setting that is relevant for communication between fault-tolerant quantum computers. In this work, we construct asymptotically good quantum codes that can be encoded and unencoded by quantum circuits of logarithmic depth and consisting of a linear total number of gates. The classical decoding algorithms also run in logarithmic depth and use $\mathcal{O}(n \log n)$ gates, or alternatively a linear number of gates but with higher depth. We further construct explicit and asymptotically good quantum codes whose encoding, unencoding and decoding all use a linear number of gates, and additionally whose encoding and unencoding may be run in logarithmic depth.Recent years have seen rapid development in the subject of quantum coding theory, with breakthroughs on many exciting classes of codes, including quantum LDPC codes, quantum locally testable codes, and quantum codes with interesting transversal gates. However, a natural class of quantum codes, which has been well-studied classically, has not yet been treated: those which can be quickly encoded and decoded. This problem concerns the channel capacity setting, where a noise channel sits between perfect encoding and unencoding/decoding operations; this is the setting that is relevant for communication between fault-tolerant quantum computers. In this work, we construct asymptotically good quantum codes that can be encoded and unencoded by quantum circuits of logarithmic depth and consisting of a linear total number of gates. The classical decoding algorithms also run in logarithmic depth and use $\mathcal{O}(n \log n)$ gates, or alternatively a linear number of gates but with higher depth. We further construct explicit and asymptotically good quantum codes whose encoding, unencoding and decoding all use a linear number of gates, and additionally whose encoding and unencoding may be run in logarithmic depth.

quant-pharXiv:2603.04543Mar 6, 20260 citations

The Code Distortion Problem

Huck Bennett, Matthew Fox, Bryant Morrell

Two linear error-correcting codes $\cal{C}_1, \cal{C}_2 \subseteq \mathbb{F}_q^n$ are called linearly equivalent if there is a linear isometry mapping $\cal{C}_1$ to $\cal{C}_2$ . In this work, we generalize the notion of linear equivalence and study the minimum distortion $\cal{D}(\cal{C}_1, \cal{C}_2)$ of a linear mapping between codes $\cal{C}_1, \cal{C}_2 \subseteq \mathbb{F}_q^n$ , which quantifies how similar $\cal{C}_1$ and $\cal{C}_2$ are. We introduce and study the Code Distortion Problem (CDP), which asks to find a minimum distortion mapping between two input codes $\cal{C}_1$ and $\cal{C}_2$ . CDP generalizes the Linear Code Equivalence Problem (LCE), which is essentially the special case of CDP where $\cal{D}(\cal{C}_1, C_2) = 1$ and which is well-studied because of its role in cryptography. We prove that (decisional) CDP is $\mathsf{NP}$ -hard to approximate to within any constant factor, and that it is in $Σ_2^P$ . We also give a single-exponential-time $k^2$ -approximation algorithm for CDP, where $k$ is the dimension of the input codes. Furthermore, we give a single-exponential-time $\big(\frac{2k + 1}{3})^2$ -approximation algorithm for a natural special case of CDP, and we show that our analysis is tight in this case. We use techniques from analogous work on the Lattice Distortion Problem (LDP) by Bennett, Dadush, and Stephens-Davidowitz (ESA, 2016). We also introduce or study a number of additional concepts that might be of independent interest. These include an adaptation of the celebrated reduction of Goldreich, Micciancio, Safra, and Seifert (IPL, 1999) from the Shortest Vector Problem (SVP) to the Closest Vector Problem (CVP) on lattices to the analogous problems on codes; successive minima bases for codes; and the matrix $0 \to 0$ "norm" on subspaces.Two linear error-correcting codes $\cal{C}_1, \cal{C}_2 \subseteq \mathbb{F}_q^n$ are called linearly equivalent if there is a linear isometry mapping $\cal{C}_1$ to $\cal{C}_2$. In this work, we generalize the notion of linear equivalence and study the minimum distortion $\cal{D}(\cal{C}_1, \cal{C}_2)$ of a linear mapping between codes $\cal{C}_1, \cal{C}_2 \subseteq \mathbb{F}_q^n$, which quantifies how similar $\cal{C}_1$ and $\cal{C}_2$ are. We introduce and study the Code Distortion Problem (CDP), which asks to find a minimum distortion mapping between two input codes $\cal{C}_1$ and $\cal{C}_2$. CDP generalizes the Linear Code Equivalence Problem (LCE), which is essentially the special case of CDP where $\cal{D}(\cal{C}_1, C_2) = 1$ and which is well-studied because of its role in cryptography. We prove that (decisional) CDP is $\mathsf{NP}$-hard to approximate to within any constant factor, and that it is in $Σ_2^P$. We also give a single-exponential-time $k^2$-approximation algorithm for CDP, where $k$ is the dimension of the input codes. Furthermore, we give a single-exponential-time $\big(\frac{2k + 1}{3})^2$-approximation algorithm for a natural special case of CDP, and we show that our analysis is tight in this case. We use techniques from analogous work on the Lattice Distortion Problem (LDP) by Bennett, Dadush, and Stephens-Davidowitz (ESA, 2016). We also introduce or study a number of additional concepts that might be of independent interest. These include an adaptation of the celebrated reduction of Goldreich, Micciancio, Safra, and Seifert (IPL, 1999) from the Shortest Vector Problem (SVP) to the Closest Vector Problem (CVP) on lattices to the analogous problems on codes; successive minima bases for codes; and the matrix $0 \to 0$ "norm" on subspaces.

arXiv:2607.26261Jul 28, 20260 citations

Distance verification for LDPC codes

Ilya I. Dumer, Alexey A. Kovalev, Leonid P. Pryadko

The problem of finding code distance has been long studied for the generic ensembles of linear codes and led to several algorithms that substantially reduce exponential complexity of this task. However, no asymptotic complexity bounds are known for distance verification in other ensembles of linear codes. Our goal is to re-design the existing generic algorithms of distance verification and derive their complexity for LDPC codes. We obtain new complexity bounds with provable performance expressed in terms of the erasure-correcting thresholds of long LDPC codes. These bounds exponentially reduce complexity estimates known for linear codes.The problem of finding code distance has been long studied for the generic ensembles of linear codes and led to several algorithms that substantially reduce exponential complexity of this task. However, no asymptotic complexity bounds are known for distance verification in other ensembles of linear codes. Our goal is to re-design the existing generic algorithms of distance verification and derive their complexity for LDPC codes. We obtain new complexity bounds with provable performance expressed in terms of the erasure-correcting thresholds of long LDPC codes. These bounds exponentially reduce complexity estimates known for linear codes.

arXiv:1605.02410IEEE International Symposium on Information Theory (ISIT)Jul 1, 20161 citation

Distance Verification for LDPC Codes

Ilya I. Dumer, Alexey A. Kovalev, Leonid P. Pryadko +1 more

The problem of finding code distance has been long studied for the generic\nensembles of linear codes and led to several algorithms that substantially\nreduce exponential complexity of this task. However, no asymptotic complexity\nbounds are known for distance verification in other ensembles of linear codes.\nOur goal is to re-design the existing generic algorithms of distance\nverification and derive their complexity for LDPC codes. We obtain new\ncomplexity bounds with provable performance expressed in terms of the\nerasure-correcting thresholds of long LDPC codes. These bounds exponentially\nreduce complexity estimates known for linear codes.The problem of finding code distance has been long studied for the generic\nensembles of linear codes and led to several algorithms that substantially\nreduce exponential complexity of this task. However, no asymptotic complexity\nbounds are known for distance verification in other ensembles of linear codes.\nOur goal is to re-design the existing generic algorithms of distance\nverification and derive their complexity for LDPC codes. We obtain new\ncomplexity bounds with provable performance expressed in terms of the\nerasure-correcting thresholds of long LDPC codes. These bounds exponentially\nreduce complexity estimates known for linear codes.

Jan 1, 20162 citations

On decoding algorithms for binary linear block codes

Bruce L. Montgomery

This dissertation investigates the complexity of minimum-distance decoding (MDD) algorithms for general binary linear block codes, and also investigates the use of such codes in constructing error correcting/detecting codes that are effective against both random and unidirectional errors. A new MDD algorithm is proposed, and its complexity is analyzed and compared to that of other such algorithms. The question of whether all good codes satisfy certain properties required for efficient implementation of the algorithm is also considered. The performance of decoding algorithms, with the information bit error rate as the criterion instead of the block error rate, is also studied. New decoding rules which can yield lower information bit error rates than the MDD rule are proposed and evaluated. A new class of binary codes is developed which corrects random errors and detects all unidirectional errors. These codes are obtained by appending a suffix to each codeword of a linear block code. To evaluate the efficiency of the new codes, a thorough comparison with the best previously obtained codes is made.This dissertation investigates the complexity of minimum-distance decoding (MDD) algorithms for general binary linear block codes, and also investigates the use of such codes in constructing error correcting/detecting codes that are effective against both random and unidirectional errors. A new MDD algorithm is proposed, and its complexity is analyzed and compared to that of other such algorithms. The question of whether all good codes satisfy certain properties required for efficient implementation of the algorithm is also considered. The performance of decoding algorithms, with the information bit error rate as the criterion instead of the block error rate, is also studied. New decoding rules which can yield lower information bit error rates than the MDD rule are proposed and evaluated. A new class of binary codes is developed which corrects random errors and detects all unidirectional errors. These codes are obtained by appending a suffix to each codeword of a linear block code. To evaluate the efficiency of the new codes, a thorough comparison with the best previously obtained codes is made.

Jan 1, 19870 citations

n

and dimension

k

over any finite field

\mathbb{F}_{q}

, we show: 1)A deterministic algorithm running in

2^{n+o(n+q)}

time for LCE. 2)A randomized algorithm running in

2^{n / 2+o(n+q)}

time for LCE and PCE. 3)A quantum algorithm running in

2^{n / 3+o(n+q)}

time for LCE and PCE. The second two algorithms improve on recent work of Nowakowski (PQCrypto 2025), which gave algorithms with similar running times but only for code equivalence on random codes and only over fields of order

q \geq 7

\mathcal{O}(n \log n)

gates, or alternatively a linear number of gates but with higher depth. We further construct explicit and asymptotically good quantum codes whose encoding, unencoding and decoding all use a linear number of gates, and additionally whose encoding and unencoding may be run in logarithmic depth.

Two linear error-correcting codes

\cal{C}_1, \cal{C}_2 \subseteq \mathbb{F}_q^n

are called linearly equivalent if there is a linear isometry mapping

\cal{C}_1

\cal{C}_2

. In this work, we generalize the notion of linear equivalence and study the minimum distortion

\cal{D}(\cal{C}_1, \cal{C}_2)

of a linear mapping between codes

\cal{C}_1, \cal{C}_2 \subseteq \mathbb{F}_q^n

, which quantifies how similar

\cal{C}_1

and

\cal{C}_2

are. We introduce and study the Code Distortion Problem (CDP), which asks to find a minimum distortion mapping between two input codes

\cal{C}_1

and

\cal{C}_2

. CDP generalizes the Linear Code Equivalence Problem (LCE), which is essentially the special case of CDP where

\cal{D}(\cal{C}_1, C_2) = 1

and which is well-studied because of its role in cryptography. We prove that (decisional) CDP is

\mathsf{NP}

-hard to approximate to within any constant factor, and that it is in

Σ_2^P

. We also give a single-exponential-time

k^2

-approximation algorithm for CDP, where

k

is the dimension of the input codes. Furthermore, we give a single-exponential-time

\big(\frac{2k + 1}{3})^2

-approximation algorithm for a natural special case of CDP, and we show that our analysis is tight in this case. We use techniques from analogous work on the Lattice Distortion Problem (LDP) by Bennett, Dadush, and Stephens-Davidowitz (ESA, 2016). We also introduce or study a number of additional concepts that might be of independent interest. These include an adaptation of the celebrated reduction of Goldreich, Micciancio, Safra, and Seifert (IPL, 1999) from the Shortest Vector Problem (SVP) to the Closest Vector Problem (CVP) on lattices to the analogous problems on codes; successive minima bases for codes; and the matrix

0 \to 0

"norm" on subspaces.