PUZZLE #3951: Hamming (7,4) Error Correction (diff 7)

hamming learn difficulty: 7/7 compute author: system unsolved

A 12-block Hamming-coded message with 10 bit errors. Each 7-bit block protects 4 data bits. Use the parity-check matrix to detect and correct single-bit errors.

DATA

Encoded	`010111001010101100001001110001001011010100010010100010110100101010010001011011110000`
Block Size	`7`
Data Bits	`4`
N Blocks	`12`
Parity Check	`[1, 1, 0, 1, 1, 0, 0], [1, 0, 1, 1, 0, 1, 0], [0, 1, 1, 1, 0, 0, 1]`
Syndrome Map	{ "(1, 1, 0)": 1, "(1, 0, 1)": 2, "(0, 1, 1)": 3, "(1, 1, 1)": 4, "(1, 0, 0)": 5, "(0, 1, 0)": 6, "(0, 0, 1)": 7 }
Errors	`10`
Hint	`Compute syndrome s = H·r for each 7-bit block. If s ≠ (0,0,0), find the column in H matching s — that's the error position. Flip that bit, extract data bits (positions 0-3), reassemble ASCII bytes`
Answer Format	`lowercase letters, no spaces or punctuation`

{
  "type": "hamming",
  "encoded": "010111001010101100001001110001001011010100010010100010110100101010010001011011110000",
  "block_size": 7,
  "data_bits": 4,
  "n_blocks": 12,
  "parity_check": [
    [
      1,
      1,
      0,
      1,
      1,
      0,
      0
    ],
    [
      1,
      0,
      1,
      1,
      0,
      1,
      0
    ],
    [
      0,
      1,
      1,
      1,
      0,
      0,
      1
    ]
  ],
  "syndrome_map": {
    "(1, 1, 0)": 1,
    "(1, 0, 1)": 2,
    "(0, 1, 1)": 3,
    "(1, 1, 1)": 4,
    "(1, 0, 0)": 5,
    "(0, 1, 0)": 6,
    "(0, 0, 1)": 7
  },
  "errors": 10,
  "hint": "Compute syndrome s = H\u00b7r for each 7-bit block. If s \u2260 (0,0,0), find the column in H matching s \u2014 that's the error position. Flip that bit, extract data bits (positions 0-3), reassemble ASCII bytes",
  "answer_format": "lowercase letters, no spaces or punctuation"
}

author's note: Pool fill: hamming diff 7