How is the syndrome for the hamming code interpreted?

The syndrome for a Hamming code is interpreted by computing the product of the received word (\mathbf{r}) and the transpose of the parity-check matrix (\mathbf{H}^T), which results in a syndrome vector. This vector is independent of the transmitted codeword and depends solely on the error pattern introduced during transmission.

Interpretation Steps:

1. Syndrome Calculation: The syndrome is calculated as (\mathbf{r}\mathbf{H}^T), where (\mathbf{r}) is the received word and (\mathbf{H}) is the parity-check matrix. This operation is essentially a linear transformation that identifies the error pattern.

2. Error Detection and Correction:

   • No Error: If the syndrome is all zeros, it indicates that no error has occurred during transmission.
   • Single Error: If the syndrome contains exactly one '1', it typically indicates that a parity bit has been flipped. However, in most cases, a single '1' in the syndrome suggests a specific error location, which can be corrected by flipping the bit at that location.
   • Multiple Errors: If more than one '1' is present in the syndrome, it suggests that multiple errors have occurred. For standard Hamming codes, if two errors occur, they cannot be corrected but can be detected.

3. Syndrome Table: For a specific Hamming code like the (7,4) code, a syndrome table is used to map each possible syndrome to the corresponding error location. This table helps in identifying which bit needs correction based on the syndrome.

In summary, the syndrome for a Hamming code is used to identify whether an error has occurred and, if so, where the error is located, allowing for single-bit error correction and detection of two-bit errors123.