What is the relationship between the distance between parity bits and their…

Concept. In the standard Hamming construction a code word position’s binary index IS that position’s column of the parity-check matrix. A set of positions can therefore serve as parity positions only while their index patterns are linearly independent over GF(2). r independent checks produce an r-bit syndrome, and an r-bit syndrome can name any one of 2r − 1 positions. What the code then does with that reach is set by the two standard bounds — detect up to t errors when dmin ≥ t + 1, correct up to t errors when dmin ≥ 2t + 1.

Application to parity-bit spacing.

1. The standard layout puts the parity bits at 1, 2, 4, 8, … — the powers of two 20, 21, 22, 23. Their indices 0001, 0010, 0100, 1000 are four unit patterns, hence four independent checks, and the gaps between them widen as 1, 2, 4, …

2. Crowd those parity bits onto the neighbouring positions 1, 2, 3, 4 instead. The indices become 0001, 0010, 0011, 0100, and 0001 XOR 0010 = 0011 — the four columns span only a three-dimensional space.

3. Parity positions must be linearly independent for the encoder to solve for the parity bits at all, so the crowded layout never delivers four working checks: at most three carry independent information, and the syndrome’s reach falls from 24 − 1 = 15 positions to 23 − 1 = 7.

4. Fewer positions the syndrome can name means fewer single-bit errors that can be located and repaired. Inside this construction, pulling the parity bits together costs capability; spreading them onto the powers of two is what preserves it.

Cross-check by counting. The same four parity bits buy very different reach depending on where they are allowed to sit:

Scope of the claim. This is a statement about the standard Hamming construction, where a position’s binary index is its column of the parity-check matrix. It is not a general law of block codes: take a finished Hamming code and permute all of its coordinates and the parity symbols can end up adjacent while the parity-check matrix keeps its rank, dmin stays 3, and the code still corrects one error. That is a relabelling of an existing code, not the construction this question describes. Bihar STET’s official key for this paper marks the closer/lower relationship as the intended answer.

Contrast with the near-miss readings.

• “Closer parity bits provide higher error correction capability” runs the relationship backwards — crowding the parity positions removes an independent check rather than adding one.

• “Distance between parity bits has no impact on error correction capability” holds only under the relabelling described above; inside the construction the parity positions are not free to move, because the dependency 0001 XOR 0010 = 0011 disqualifies the crowded set.

• “The position of parity bits is irrelevant in Hamming Code” is the strongest form of that same reading; the powers-of-two positions are precisely what let the failed checks spell out the binary index of the damaged bit.

Result. Closer parity bits provide lower error correction capability. The powers-of-two spacing keeps the r checks independent, which maximises the syndrome’s reach of 2r − 1 positions and, with the number of parity bits fixed for m data bits by 2r ≥ m + r + 1, gives the code its full detection and correction strength.