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QR codes use Reed-Solomon error correction over the finite field F 256 {\displaystyle \mathbb {F} _{256}} or GF(28), the elements of which are encoded as bytes of 8 bits.

QR codes utilize Reed-Solomon error correction to ensure data integrity and allow for data recovery in cases where the QR code is partially damaged or obscured.

The Reed-Solomon code uses one of 37 different polynomials over F 256 {\displaystyle \mathbb {F} _{256}}, with degrees ranging from 7 to 68, depending on how many error correction bytes the code adds.

Here's a detailed breakdown of the error correction mechanism in QR codes:

Reed-Solomon Error Correction

Finite Field F256\mathbb{F}_{256}F256 or GF(28)

1.Elements as Bytes:

The elements of the finite field F256\mathbb{F}_{256}F256 are encoded as 8-bit bytes. A byte b7b6b5b4b3b2b1b0b_7 b_6 b_5 b_4 b_3 b_2 b_1 b_0b7b6b5b4b3b2b1b0 represents the binary form where bib_ibi (for iii from 0 to 7) are the bits.

The numerical value of the byte is ∑i=07bi2i\sum_{i=0}^{7} b_i 2^i∑i=07bi2i.

2.Field Elements:

Each byte encodes a field element ∑i=07biαi\sum_{i=0}^{7} b_i \alpha^i∑i=07biαi, where α\alphaα is a primitive element in F256\mathbb{F}_{256}F256.

The primitive element α\alphaα satisfies the polynomial equation α8+α4+α3+α2+1=0\alpha^8 + \alpha^4 + \alpha^3 + \alpha^2 + 1 = 0α8+α4+α3+α2+1=0.

The corresponding polynomial for α\alphaα is x8+x4+x3+x2+1x^8 + x^4 + x^3 + x^2 + 1x8+x4+x3+x2+1, which is polynomial number 285, with an initial root of 0.

Error Correction Code Structure

1.Error Correction Polynomials:

The Reed-Solomon code in QR codes uses 37 different polynomials over F256\mathbb{F}_{256}F256, depending on the error correction level and the QR code version.

These polynomials are of varying degrees (from 7 to 68), depending on the number of error correction bytes required.

2.Systematic BCH View:

QR codes use a systematic BCH view of the Reed-Solomon code.

The generator polynomial for error correction is typically of the form ∏i=0n-1(x-αi)\prod_{i=0}^{n-1}(x - \alpha^i)∏i=0n-1(x-αi), where nnn is the degree of the polynomial and is specific to the QR code standard.

Error Correction Levels

QR codes have four error correction levels, each providing different amounts of error protection:

1.Level L (Low): Recovers up to 7% of the data.

2.Level M (Medium): Recovers up to 15% of the data.

3.Level Q (Quartile): Recovers up to 25% of the data.

4.Level H (High): Recovers up to 30% of the data.

Error Correction Process

1.Encoding:

During encoding, error correction codes are generated and appended to the original data.

The number of error correction bytes added depends on the chosen error correction level and the QR code version (size).

2.Decoding and Error Correction:

When scanning a QR code, the reader uses the error correction bytes to detect and correct errors.

The process involves polynomial division in F256\mathbb{F}_{256}F256 to locate and correct errors in the received data.

3.Error Detection:

The Reed-Solomon algorithm can detect errors and erasures (unknown locations of errors).

It uses the properties of polynomials over F256\mathbb{F}_{256}F256 to reconstruct the original message even if parts of the data are corrupted or missing.

Practical Example

1.Generator Polynomial:

Suppose we need a generator polynomial for a QR code with error correction capacity ttt. The generator polynomial might be g(x)=(x-α0)(x-α1)-(x-α2t-1)g(x) = (x - \alpha^0)(x - \alpha^1) \cdots (x - \alpha^{2t-1})g(x)=(x-α0)(x-α1)-(x-α2t-1).

2.Encoding Data:

The original data is treated as a polynomial m(x)m(x)m(x).

The error correction codeword is computed by multiplying m(x)m(x)m(x) by x2tx^{2t}x2t (shifting it by 2t2t2t places) and then taking the remainder when divided by the generator polynomial g(x)g(x)g(x).

3.Decoding Data:

The received polynomial r(x)r(x)r(x) is divided by the generator polynomial g(x)g(x)g(x) to detect and correct errors.

If errors are detected, the algorithm determines the error locations and values to correct them.

By leveraging the robust Reed-Solomon error correction mechanism, QR codes can reliably transmit data even in the presence of significant noise or damage. This makes them highly reliable for a wide range of applications, from commercial product tracking to information sharing.

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