How can checks locate a one-bit error?
A bit can flip while a string of 0s and 1s is being sent. Extra checking bits help the receiver notice something is wrong.
Here, overlapping checks give each position a different combination. If at most one bit is wrong, the failed checks identify its position. Flip it back and the original message is recovered.
Now try two wrong bits and watch the same method make a misleading diagnosis. Error correction has limits. An extra check distinguishes more cases, but still cannot recover arbitrary damage.
Make a wrong repair, then let the extended code stop it
- Flip one bit. Can the checks locate and repair it?
- Flip two bits. Watch why the ordinary correction rule can give a wrong answer.
- Switch to the extended code with the same errors. Reporting a double error does not mean locating both wrong bits.
Matrices, distance and the limit of the promise
Further controls and numerical checks
Enter exploration, select a four-bit message and code, then toggle channel flips by position. Each button shows its fixed position number and current received value. Tab moves keyboard focus; Space or Enter toggles a bit. Presets select no flips, a single flip at 6, or double flips at 3 and 5. Reset restores message 1011, the extended code, and a flip at 6. The shared player provides pause, replay and chapter seeking.
With ordinary Hamming selected, flip positions 3 and 5. Their combined syndrome is 3 XOR 5 = 6. The single-error decoder wrongly flips position 6, yielding 0100101 and payload 0101. That is a valid codeword for a different message. In a detection-only policy, a nonzero Hamming syndrome can flag every double error. The problem is that this policy cannot simultaneously distinguish double errors from the single errors it is supposed to correct.
Switch to extended Hamming(8,4). An eighth bit makes the whole word even. The comparison preserves the four data bits and the same channel positions, 3 and 5. Let s be the original three-check syndrome and q the XOR of all eight received bits:
| Received checks | Assuming at most two flips | Decoder action |
|---|---|---|
| s=0, q=0 | No error detected | Accept |
| s≠0, q=1 | One error in the first seven positions | Flip position s |
| s=0, q=1 | Error in the added overall check bit | Flip position 8 |
| s≠0, q=0 | Double error | Flag; withhold data; do not correct |
Either flipped position can be a data bit, an ordinary check bit, or the added eighth bit. Detecting a double error does not identify the two damaged positions: several original codewords remain equally plausible.
Want to check the technical details?
A bit may flip while a sequence of zeros and ones is being transmitted. We add check bits and inspect overlapping groups. If just one bit is wrong, the combination of failed checks can reveal its position. More errors can defeat that promise.
A single overall parity bit can expose one flip, but cannot locate it. Hamming(7,4) keeps four data bits at positions 3, 5, 6, 7 and adds check bits at 1, 2, 4. This scene uses even parity: each group must contain an even number of ones, or equivalently, its bits must XOR to zero.
The groups are C1 = {1,3,5,7}, C2 = {2,3,6,7}, and C4 = {4,5,6,7}. Their overlap is deliberate. Every position belongs to a different combination of checks. Position 6, written as binary 110, belongs to C4 and C2 but not C1. The small numbers identify positions; the zero or one inside each circle is the bit’s value.
The film follows the message 1011, encoded as 0110011. A channel flip at position 6 produces 0110001. Rechecking gives C1=0, C2=1, C4=1. Read in the order C4, C2, C1, the syndrome is 110₂ = 6. Assuming at most one error, flip that position back and extract positions 3, 5, 6, 7 to recover 1011. The decoder uses only the received word. The original data remains visible as a teaching comparison, not as an input to decoding.
Columns below are positions 1 through 7. Rows of the generator matrix correspond to data bits d1 through d4. The model calculates c = dG over the binary field GF(2), where addition is XOR. Data columns 3, 5, 6, 7 form an identity matrix, making this a systematic code: the message bits remain directly extractable.
G = 1 1 1 0 0 0 0
1 0 0 1 1 0 0
0 1 0 1 0 1 0
1 1 0 1 0 0 1
H = 1 0 1 0 1 0 1 C1
0 1 1 0 0 1 1 C2
0 0 0 1 1 1 1 C4The matrices satisfy HGᵀ = 0. For a received column vector r, Hr gives the three check results. The extended encoder appends the XOR of the first seven bits at position 8. Its check matrix adds a zero column to H and a final row of eight ones.
Hamming distance counts positions where two equal-length words differ. Parity is the XOR of a selected group of bits. These are different quantities. Exhaustively comparing all 16 valid codewords gives minimum distance 3 for the ordinary code and 4 for the extended code. The SECDED decoding rule used here guarantees single-error correction and double-error detection, not double-error correction.
Exploration allows more than two flips so you can test the boundary. In the extended code, flips at 1, 2, 3 masquerade as an error at position 8. Flips at 1, 2, 3, 8 produce a different valid word and go undetected. “No error detected” is therefore not a universal guarantee that the original message survived. This is a complete small binary block-code model, not a particular memory device’s full ECC system, retransmission protocol or statistical fault model.
See Hamming’s original 1950 paper, sections 3–5 for the construction and extension, MIT 6.02’s error-correction lecture for distance and correction capability, and the MIT 6.111 tutorial answers for double-error and overall-parity reasoning. The example words and counterexamples here are computed by the independent scene model.