⚙️ Hamming Codes

Code variant

Data bits (k)4

Parity bits (r)3

Codeword bits (n)7

Code rate4/7 ≈ 0.571

Min distance (d_min)3

Errors detected1

Errors corrected1

Hamming (1950)
Parity bits at positions 2⁰, 2¹, 2², …
Each covers a subset of data bits.
Syndrome s = H·r gives error position.
SECDED adds overall parity for d_min = 4.

① Message bits (click to toggle)

② Encoded codeword (click any bit to flip — simulate an error)

✓ No errors detected

③ Corrected / received message bits

Parity-check matrix H (rows = parity checks)

Syndrome s = H · r̃ (received codeword)

s = 000 → no error