QR Code Data Encoder/Decoder

In numeric mode, groups of three decimal digits are converted to a 10-bit binary number (0–999 fits in 10 bits). If the digit count is not divisible by three, the remaining one or two digits are encoded in 4 or 7 bits respectively. The mode indicator for numeric mode is 0001 (4 bits), followed by a character count indicator whose length varies by version (10 bits for Version 1–9, 12 for Version 10–26, 14 for Version 27–40). This encoding is approximately 3.32 bits per digit, compared to 8 bits per character in byte mode, making it highly efficient for numeric-only payloads such as tracking numbers, serial numbers, or phone numbers.

Reed-Solomon (RS) codes are a class of polynomial error-correcting codes invented by Irving Reed and Gustave Solomon in 1960. In QR codes, RS is applied to groups of 8-bit data codewords, adding redundant error-correction codewords computed from Galois Field GF(256) arithmetic. The number of error-correction codewords per block is determined by the version and error-correction level. Each RS block can detect and correct up to t errors where 2t equals the number of error-correction codewords; beyond this capacity, the codewords can still detect but not correct errors. Multiple RS blocks are interleaved before placement in the symbol so that a physical burst error (such as a scratch) affects multiple blocks, reducing the probability that any single block exceeds its correction capacity.

The eight masking patterns defined in ISO/IEC 18004 are XOR functions applied to the data modules of the QR code to prevent large areas of uniform dark or light modules, isolated modules, or patterns that resemble the finder patterns from appearing in the symbol. Such visual anomalies cause decoding errors in some scanner optical systems. Each of the eight mask patterns is applied in turn, and the resulting symbol is scored using a four-penalty rule system that assigns penalty points for runs, boxes, and finder-pattern-like sequences. The mask with the lowest total penalty score is selected and its index encoded in the format information strip. Different encoder implementations may choose different masks for the same data, which is why two encoders can produce visually different QR codes from identical input.

The quiet zone is not encoded in bits — it is a physical border of blank (light) modules around the outside of the QR symbol, not part of the data region. Its purpose is to provide a contrast boundary that allows the scanner's image processor to locate the edges of the symbol against the background. ISO/IEC 18004 requires a minimum quiet zone of four module widths on all sides. The quiet zone carries no data and is not subject to error correction; it is simply part of the physical printing specification. Errors in the quiet zone (such as background pattern encroaching on the margin) are detected at the scanner hardware level before the RS decoder is invoked.

Kanji mode uses 13-bit encoding for double-byte characters in the Shift-JIS character set, specifically the ranges 0x8140–0x9FFC and 0xE040–0xEBBF. The encoding subtracts a base value from each two-byte character and packs the result into 13 bits, achieving roughly 8 bits per character for standard Japanese kanji and kana — more efficient than encoding the same characters as UTF-8 (which requires 3 bytes = 24 bits per character in byte mode). However, Kanji mode only covers Shift-JIS encoded characters; emojis, Simplified Chinese characters, and other Unicode outside the Shift-JIS range must use byte mode with UTF-8. For modern multilingual content, byte mode with UTF-8 is more universal, though less space-efficient for pure Japanese text.