Hamming1511

Hamming1511, commonly written as Hamming(15,11), is a binary linear error-correcting code with block length 15 and dimension 11. It belongs to the family of Hamming codes introduced by Richard W. Hamming. The code has 2^11 = 2048 codewords of length 15, and its redundancy is 4, giving a code rate of 11/15 ≈ 0.733.

The code’s minimum distance is 3, enabling correction of any single-bit error and detection of most two-bit errors. Hamming(15,11) is a perfect single-error-correcting code, meaning the Hamming spheres of radius 1 around codewords partition the entire space of length-15 binary vectors.

Construction typically uses a parity-check matrix H of size 4x15 containing every nonzero 4-bit binary vector as a column. Parity bits are usually placed at bit positions that are powers of two (1, 2, 4, 8), with data bits occupying the remaining positions (3,5,6,7,9,10,11,12,13,14,15). A generator matrix G in systematic form can be formed as [I_11 | P], and codewords are obtained by c = mG for an 11-bit message m.

Decoding proceeds by computing the syndrome s = Hc^T. If s is the zero vector, the received word is considered error-free. If s is nonzero, it corresponds to the binary index of the erroneous bit, which is then flipped to recover the nearest codeword. For Hamming(15,11), single-bit errors are corrected; two-bit errors may be detected but are not reliably corrected without extending the code.

Hamming(15,11) serves as a foundational example in coding theory and informs extended Hamming codes and SECDED schemes, such as the extended Hamming code (16,11) that adds an overall parity bit for double-error detection.

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