Error Detection and Correction

CRC codes and block and convolutional channel coding

Apply error detection and correction using:

Cyclic redundancy check (CRC) coding techniques. CRC generators and detectors that are optimized for HDL are also included.
Block coding techniques including linear block codes, Hamming codes, Bose–Chaudhuri–Hocquenghem (BCH) codes, Reed-Solomon (RS) codes, low-density parity-check (LDPC) codes, and turbo product codes (TPC). RS encoders and decoders that are optimized for HDL are also included.
Convolutional coding techniques including convolutional, turbo, a posteriori probability (APP), and Viterbi techniques. Turbo and Viterbi decoders that run with GPUs for processing speed improvement are also included.
Galois fields to enable manipulation of even and odd ordered elements of finite fields.

To learn more about error control codes, see Error Detection and Correction.

Functions

Cyclic Redundancy Check Codes

`crcConfig`	Cyclic redundancy check (CRC) coding configuration (Since R2024a)
`crcGenerate`	Generate CRC code bits and append them to input data (Since R2024a)
`crcDetect`	Detect errors in input data using CRC parity bits (Since R2024a)

Block Coding

Linear Block Codes

`cyclgen`	Produce parity-check and generator matrices for cyclic code
`cyclpoly`	Produce generator polynomials for binary cyclic code
`decode`	Block decoder
`encode`	Block encoder
`gfweight`	Calculate minimum distance of linear block code
`gen2par`	Convert between parity-check and generator matrices
`hammgen`	Parity-check and generator matrices for Hamming code
`syndtable`	Produce syndrome decoding table

BCH Codes

`bchenc`	BCH encoder
`bchdec`	BCH decoder
`bchgenpoly`	Produce generator polynomials for BCH code
`bchnumerr`	Number of correctable errors for BCH code

Reed-Solomon Codes

`rsenc`	Reed-Solomon encoder
`rsdec`	Reed-Solomon decoder
`rsgenpoly`	Generator polynomial of Reed-Solomon code
`rsgenpolycoeffs`	Generator polynomial coefficients of Reed-Solomon code

LDPC Codes

`ldpcEncode`	Encode binary LDPC code (Since R2021b)
`ldpcDecode`	Decode binary LDPC code (Since R2021b)
`ldpcEncoderConfig`	Create LDPC encoder configuration (Since R2021b)
`ldpcDecoderConfig`	Create LDPC decoder configuration (Since R2021b)
`ldpcQuasiCyclicMatrix`	Parity-check matrix of quasi-cyclic LDPC code (Since R2021b)
`dvbs2ldpc`	Low-density parity-check (LDPC) codes from DVB-S.2 standard

Turbo Product Codes

`tpcenc`	Turbo product code (TPC) encoder
`tpcdec`	Turbo product code (TPC) decoder

Convolutional Coding

`convenc`	Convolutionally encode binary message
`vitdec`	Convolutionally decode binary data by using Viterbi algorithm
`distspec`	Compute distance spectrum of convolutional code
`getTurboIOIndices`	Compute output indices for turbo coding (Since R2021a)
`iscatastrophic`	True for trellis corresponding to catastrophic convolutional code
`istrellis`	True for valid trellis structure
`oct2dec`	Convert octal to decimal numbers
`poly2trellis`	Convert convolutional code polynomials to trellis description

Galois Fields

Galois Field Computations — Even Order Finite Fields

`convmtx`	Convolution matrix of Galois field vector
`cosets`	Produce cyclotomic cosets for Galois field
`dftmtx`	Discrete Fourier transform matrix in Galois field
`fft`	Fast Fourier transform of Galois field vector
`filter (gf)`	1-D digital filter over Galois field
`gf`	Galois field array
`gftable`	Generate file to accelerate Galois field computations
`ifft`	Inverse fast Fourier transform of Galois field vector
`isprimitive`	True for primitive polynomial for Galois field
`log`	Logarithm in Galois field
`minpol`	Find minimal polynomial of Galois field element
`mldivide`	Solve systems of linear equations Ax = B for x when A is a Galois array
`primpoly`	Find primitive polynomials for Galois field

Galois Fields of Odd Characteristic — Odd Order Finite Fields

`gfadd`	Add polynomials over Galois field
`gfconv`	Multiply polynomials over Galois field
`gfcosets`	Produce cyclotomic cosets for Galois field
`gfdeconv`	Divide polynomials over Galois field
`gfdiv`	Divide elements of Galois field
`gffilter`	Filter data using polynomials over prime Galois field
`gflineq`	Find particular solution of `Ax` = `b` over prime Galois field
`gfminpol`	Find minimal polynomial of Galois field element
`gfmul`	Multiply elements of Galois field
`gfpretty`	Polynomial in traditional format
`gfprimck`	Check whether polynomial over Galois field is primitive
`gfprimdf`	Provide default primitive polynomials for Galois field
`gfprimfd`	Find primitive polynomials for Galois field
`gfrank`	Compute rank of matrix over Galois field
`gfrepcov`	Convert one binary polynomial representation to another
`gfroots`	Find roots of polynomial over prime Galois field
`gfsub`	Subtract polynomials over Galois field
`gftrunc`	Minimize length of polynomial representation
`gftuple`	Simplify or convert Galois field element formatting

Objects

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Cyclic Redundancy Check Codes

`comm.CRCGenerator`	Generate CRC code bits and append to input data
`comm.CRCDetector`	Detect errors in input data using CRC
`comm.HDLCRCGenerator`	Generate CRC code bits and append to input data
`comm.HDLCRCDetector`	Detect errors in input data using CRC

Block Coding

BCH Codes

`comm.BCHEncoder`	Encode data using BCH encoder
`comm.BCHDecoder`	Decode data using BCH decoder

Reed-Solomon Codes

`comm.RSEncoder`	Encode data using Reed-Solomon encoder
`comm.RSDecoder`	Decode data using Reed-Solomon decoder
`comm.HDLRSEncoder`	Encode message using Reed-Solomon encoder
`comm.HDLRSDecoder`	Decode message using Reed-Solomon decoder

Convolutional Coding

`comm.ConvolutionalEncoder`	Convolutionally encode binary data
`comm.gpu.ConvolutionalEncoder`	Convolutionally encode binary data with GPU
`comm.TurboEncoder`	Encode input signal using turbo encoder parallel concatenated encoding scheme
`comm.APPDecoder`	Decode convolutional code by using APP method
`comm.TurboDecoder`	Decode input signal using turbo decoder parallel concatenated decoding scheme
`comm.gpu.TurboDecoder`	Decode input signal using turbo decoding with GPU
`comm.ViterbiDecoder`	Decode convolutionally encoded data using Viterbi algorithm
`comm.gpu.ViterbiDecoder`	Decode convolutionally encoded data using Viterbi algorithm with GPU

Blocks

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Cyclic Redundancy Check Codes

General CRC Generator	Generate CRC code bits according to generator polynomial and append to input data frames
General CRC Syndrome Detector	Detect errors in received codeword frames according to generator polynomial
General CRC Generator HDL Optimized	Generate CRC code bits and append them to input data
General CRC Syndrome Detector HDL Optimized	Detect errors in input data using CRC

Block Coding

Linear Block Codes

Binary Cyclic Encoder	Create systematic cyclic code from binary vector data
Binary Cyclic Decoder	Decode systematic cyclic code to recover binary vector data
Binary Linear Encoder	Create linear block code from binary vector data
Binary Linear Decoder	Decode linear block code to recover binary vector data
Hamming Encoder	Create Hamming code from binary vector data
Hamming Decoder	Decode Hamming code to recover binary vector data

BCH Codes

BCH Encoder	Create BCH code from binary vector data
BCH Decoder	Decode BCH code to recover binary vector data

Reed-Solomon Codes

Binary-Input RS Encoder	Create Reed-Solomon code from binary vector data
Binary-Output RS Decoder	Decode Reed-Solomon code to recover binary vector data
Integer-Input RS Encoder	Create Reed-Solomon code from integer vector data
Integer-Output RS Decoder	Decode Reed-Solomon code to recover integer vector data
Integer-Input RS Encoder HDL Optimized	Encode data using a Reed-Solomon (RS) encoder
Integer-Output RS Decoder HDL Optimized	Decode data using Reed-Solomon (RS) decoder

LDPC Codes

LDPC Encoder	Encode binary low-density parity-check (LDPC) code
LDPC Decoder	Decode binary low-density parity-check (LDPC) code

Turbo Product Codes

TPC Encoder	Turbo product code (TPC) encoder
TPC Decoder	Turbo product code (TPC) decoder

Convolutional Coding

Convolutional Encoder	Encode binary data using convolutional encoding scheme
Turbo Encoder	Encode binary data using parallel concatenated encoding scheme
APP Decoder	Decode convolutional code using a posteriori probability (APP) method
Turbo Decoder	Decode input signal using parallel concatenated decoding scheme
Viterbi Decoder	Decode convolutionally encoded data using Viterbi algorithm

Topics

Error Detection and Correction
Detect errors using cyclic redundancy check (CRC) codes. Learn about various error-control channel coding techniques.
Transmit and Receive Shortened Reed-Solomon Codes
Transmit and receive standard and shortened RS-encoded, 64-QAM-modulated data through an AWGN channel.
Reed-Solomon Coding with Erasures, Punctures, and Shortening in Simulink
Configure Reed-Solomon (RS) codes to perform block coding with erasures, punctures, and shortening.
Creation, Validation, and Testing of User Defined Trellis Structure
Use MATLAB^® to create and validate a user defined trellis structure, then use a unit test bench built in Simulink^® to test the implementation.
Working with Galois Fields
This example shows how to work with Galois fields.
Representation of Polynomials in Communications Toolbox
You can specify polynomials as a character vector or string scalar by using a variety of syntaxes.

Featured Examples

DVB-S.2 Link, Including LDPC Coding

The application of low density parity check (LDPC) codes in the second generation Digital Video Broadcasting standard (DVB-S.2), which is deployed by DIRECTV in the United States. The example uses communications System objects to simulate a transmitter-receiver chain that includes LDPC encoding and decoding.

Open Script

DVB-S.2 Link, Including LDPC Coding in Simulink

The state-of-the-art channel coding scheme used in the second generation Digital Video Broadcasting standard (DVB-S.2), which is deployed by DIRECTV in the United States. The coding scheme is based on concatenation of LDPC (Low-Density Parity-Check) and BCH codes. LDPC codes, invented by Gallager in his seminal doctoral thesis in 1960, can achieve extremely low error rates near channel capacity by using a low-complexity iterative decoding algorithm. The outer BCH codes are used to correct sporadic errors made by the LDPC decoder.

Open Model

LDPC Link Simulation Using GPU Processing

Use the ldpcDecode function and gpuArray to increase the speed of a communications system simulation.

Open Live Script

Parallel Concatenated Convolutional Coding: Turbo Codes

Characterize the performance of turbo codes over a noisy channel.

Open Model

Tail-Biting Convolutional Coding

Use the Convolutional Encoder and Viterbi Decoder blocks to simulate a tail-biting convolutional code.

Open Script

Rate 2/3 Convolutional Code in AWGN

Generate a bit error rate versus Eb/No curve for a link that uses 16-QAM modulation and a rate 2/3 convolutional code in AWGN.

Open Live Script

Punctured Convolutional Coding

Use the convolutional encoder and Viterbi decoder System objects to simulate the bit error rate (BER) of a punctured coding system.

Open Live Script

Estimate BER for Hard and Soft Decision Viterbi Decoding

Estimate bit error rate (BER) performance for hard-decision and soft-decision Viterbi decoders in AWGN. Compare the performance to that of an uncoded 64-QAM link.

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Using GPUs to Accelerate Turbo Coding Bit Error Rate Simulations

Use GPUs to dramatically accelerate bit error rate simulations.

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ElGamal Public Key Cryptosystem

Use the Galois field array function, gf, to implement an ElGamal public key cryptosystem.

Open Live Script

5G LDPC Block Error Rate Simulation Using the Cloud or a Cluster

Use the cloud or a cluster for block error rate (BLER) simulation of low-density parity-check (LDPC) coding for the 5G NR downlink shared transport channel (DL-SCH).

Open Script