Will QR Codes Ever Run Out? The Astronomical Numbers Behind Their Longevity

QR code combinatorial capacity

A QR code consists of a square matrix of black‑and‑white modules. For the common 25 × 25 matrix used by WeChat payment codes, the total number of modules is 25 × 25 = 625. After reserving modules for finder patterns, timing patterns, alignment patterns and error‑correction codewords, roughly 478 modules remain available for data encoding. Each of those modules can be either black or white, giving a theoretical space of 2^478 distinct patterns.

Using the logarithmic identity log10(2^n) = n·log10(2), the decimal magnitude is:

log10(2^478) = 478 × log10(2) ≈ 478 × 0.30103 ≈ 143.8

Thus the number of possible codes is about 10^143.8 ≈ 3.8 × 10^143.

Exhaustion time under realistic usage

If the world scans 100 billion QR codes per day, the annual usage is:

100 × 10^9 scans/day × 365 days ≈ 3.65 × 10^13 scans/year

The time required to consume the entire 25 × 25 space is:

2^478 / (3.65 × 10^13) ≈ 2.14 × 10^131 years

The age of the observable universe is roughly 1.37 × 10^10 years, so the QR‑code space would not be exhausted for a period many orders of magnitude longer than the universe’s lifetime.

Logarithmic calculation method

Because the raw numbers are astronomically large, the calculation is performed in logarithmic space:

Compute log10(2^478) = 143.8.

Convert the daily scan volume to a yearly total and take its base‑10 logarithm: log10(3.65 × 10^13) ≈ 13.56.

Subtract to obtain the logarithm of the remaining years: 143.8 − 13.56 ≈ 130.24, which corresponds to 10^130.24 ≈ 2.14 × 10^131 years.

The same logarithmic approach is used for larger matrices where direct exponentiation is infeasible.

Capacity of higher‑version QR codes

QR codes have 40 standardized versions. The side length (in modules) of version v is: size(v) = 21 + 4 × (v − 1) Examples:

Version 6 (36 × 36 matrix) : 36 × 36 = 1 296 modules. After subtracting overhead, the data‑module count is still on the order of a thousand, yielding a combinatorial space of roughly 2^1200 ≈ 10^361 possible codes.

Version 40 (177 × 177 matrix) : 177 × 177 = 31 329 modules. Even after accounting for required patterns, the remaining data capacity exceeds 20 000 bits, giving a space larger than 10^20000 distinct codes.

These figures illustrate that every QR‑code version provides a search space far beyond any conceivable real‑world usage.

Technical overview of QR codes

A QR (Quick Response) code is a two‑dimensional barcode that encodes data in a grid of black and white squares. Compared with one‑dimensional barcodes, QR codes can store alphanumeric text, binary data, URLs, and even small images. The encoding process includes:

Data analysis and mode selection (numeric, alphanumeric, byte, kanji).

Error‑correction coding using Reed‑Solomon codes (levels L, M, Q, H).

Structure final message, interleaving, and placement of data bits into the matrix.

Adding functional patterns (finder, alignment, timing, version information) that are fixed for each version.

The resulting matrix can be read by any camera‑based scanner that implements the ISO/IEC 18004 standard.

Illustrative calculations (images)