Encode HELLO WORLD by hand — a complete worked example

The setup

The traditional exercise: encode the string HELLO WORLD at error-correction level Q. Every character is in the 45-character alphanumeric set, and 11 alphanumeric characters fit comfortably in version 1-Q (capacity: 16), so the target is a 21 × 21 symbol with 13 data codewords and 13 error-correction codewords in a single block. Every value below is real — you can check each stage against a decoder at the end.

Step 1: mode and count

• Mode indicator, alphanumeric: 0010
• Character count, 11, in 9 bits (versions 1–9): 000001011

Step 2: encode the characters

Alphanumeric values: H=17, E=14, L=21, O=24, space=36, W=32, R=27, D=13. Characters pair up as 45 × first + second, 11 bits per pair; the odd final character takes 6 bits:

Pair	Calculation	Value	Bits
H,E	17 × 45 + 14	779	01100001011
L,L	21 × 45 + 21	966	01111000110
O,␣	24 × 45 + 36	1116	10001011100
W,O	32 × 45 + 24	1464	10110111000
R,L	27 × 45 + 21	1236	10011010100
D	13	13	001101

Running total: 4 + 9 + 5 × 11 + 6 = 74 bits.

Step 3: terminator and padding

Version 1-Q holds 13 data codewords = 104 bits, so there is room for the full 4-bit terminator 0000 (→ 78 bits), then two 0s to reach the 80-bit byte boundary, then the alternating pad bytes 11101100 00010001 11101100 to fill 104. Slicing the stream into bytes gives the 13 data codewords:

20 5B 0B 78 D1 72 DC 4D 43 40 EC 11 EC

(In decimal: 32, 91, 11, 120, 209, 114, 220, 77, 67, 64, 236, 17, 236.)

Step 4: Reed–Solomon check codewords

Treat the 13 data codewords as polynomial coefficients and divide by the degree-13 generator polynomial over GF(256), as described in Reed–Solomon error correction. The remainder is the 13 check codewords:

A8 48 16 52 D9 36 9C 00 2E 0F B4 7A 10

(Decimal: 168, 72, 22, 82, 217, 54, 156, 0, 46, 15, 180, 122, 16.) The division is mechanical but tedious by hand — 13 rounds of XOR-and-shift using log tables for the GF(256) multiplications. Version 1-Q is a single block, so there is no interleaving: the final sequence is simply data then checks, 26 codewords, 208 bits. Version 1 adds no remainder bits.

Step 5: place, mask, finish

The 208 bits flow into the symbol in the zig-zag pattern — two-module columns from the bottom-right, up then down, skipping function patterns. Then the eight mask patterns are each applied and scored on the four penalty rules; the winner for this payload varies by implementation detail, and any choice is valid. Finally the 15-bit format information for level Q and the chosen mask is BCH-coded, XORed with 101010000010010, and written twice.

That is a complete QR code, by hand. Generate HELLO WORLD at level Q with the text tool and you get the same 21 × 21 symbol this arithmetic produces — decode it and the 26 codewords above come back out.

FAQ

Why is HELLO WORLD the standard QR encoding example?

It is short enough to fit version 1, uses the alphanumeric mode's pairing arithmetic including a space and an odd trailing character, and exercises the terminator and both pad bytes — every interesting encoding rule in one small payload.

What are the data codewords for HELLO WORLD at version 1-Q?

In hex: 20 5B 0B 78 D1 72 DC 4D 43 40 EC 11 EC. The first ten bytes carry the mode header and character data; the last three are the standard pad bytes filling the version's 13-codeword capacity.

How are the error-correction codewords calculated?

By dividing the data codeword polynomial by the degree-13 generator polynomial over GF(256) and taking the remainder: A8 48 16 52 D9 36 9C 00 2E 0F B4 7A 10. Each multiplication uses the field's log and antilog tables.

Which mask does HELLO WORLD use?

Whichever of the eight scores lowest under the encoder's penalty evaluation — implementations can legitimately differ on this, and every choice produces a valid, decodable symbol. The chosen mask number is recorded in the format information.