Hamming code - theory of information and encoding

Hamming-code.ipynb

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   "cell_type": "code",
   "execution_count": 1,
   "id": "happy-cuisine",
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   "source": [
    "from functools import reduce\n",
    "import operator as op"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "signed-bowling",
   "metadata": {},
   "source": [
    "### Hamming code class"
   ]
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   "source": [
    "class HammingCode:\n",
    "\n",
    "    @staticmethod\n",
    "    def pow_two(max):\n",
    "        n = 0.5\n",
    "        while (n := n * 2) <= max:\n",
    "            yield int(n)\n",
    "\n",
    "    @staticmethod\n",
    "    def dispatch(bits):\n",
    "        bits = bits.copy()\n",
    "        err = reduce(op.xor, (n for n, bit in enumerate(bits, 1) if bit))\n",
    "        bits[err - 1] = int(not bits[err - 1])\n",
    "        return bits\n",
    "\n",
    "    @staticmethod\n",
    "    def encode(bits):\n",
    "        bits = bits.copy()\n",
    "        for n in HammingCode.pow_two(len(bits)):\n",
    "            bits.insert(n - 1, 0)\n",
    "\n",
    "        for p in HammingCode.pow_two(len(bits)):\n",
    "            bits[p-1] = reduce(op.xor, (bit for n, bit in enumerate(bits, 1) if (n // p) % 2))\n",
    "        return bits\n",
    "\n",
    "    @staticmethod\n",
    "    def decode(bits):\n",
    "        bits = HammingCode.dispatch(bits)\n",
    "        return [bit for n, bit in enumerate(bits, 1) if (n & (n - 1)) != 0]\n"
   ]
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   "cell_type": "markdown",
   "id": "naked-identity",
   "metadata": {},
   "source": [
    "`(n // p)` gives the number of block in line and `& 2` checks if it is a pair block   \n",
    "`num & (num - 1)` does a bitwise check, like 1000 & 0100 = 0000, and 1001 & 1000 = 1000\n",
    "\n",
    "$ \\begin{pmatrix}\n",
    "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\\\  \n",
    "0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1\\\\   \n",
    "0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1\\\\    \n",
    "1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\\\\n",
    "\\end{pmatrix} $"
   ]
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   "cell_type": "markdown",
   "id": "mineral-petroleum",
   "metadata": {},
   "source": [
    "### Tests"
   ]
  },
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   "cell_type": "code",
   "execution_count": 3,
   "id": "recognized-satellite",
   "metadata": {},
   "outputs": [],
   "source": [
    "A_vector = [1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0]\n",
    "X_vector = [0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0]\n",
    "Y_vector = [0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0]\n",
    "A = [1, 1, 0, 0]\n",
    "X = [0, 1, 1, 1, 1, 0, 0]"
   ]
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   "cell_type": "code",
   "execution_count": 4,
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     "name": "stdout",
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     "text": [
      "decoded message with error: [1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0]\n",
      "original message          : [1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0]\n"
     ]
    }
   ],
   "source": [
    "print(f\"decoded message with error: {HammingCode.decode(Y_vector)}\")\n",
    "print(f\"original message          : {A_vector}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "middle-title",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "encoded message by python: [0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0]\n",
      "encoded message by matrix: [0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0]\n"
     ]
    }
   ],
   "source": [
    "print(f\"encoded message by python: {HammingCode.encode(A_vector)}\")\n",
    "print(f\"encoded message by matrix: {X_vector}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "chronic-belize",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "encoded message by python: [0, 1, 1, 1, 1, 0, 0]\n",
      "encoded message by matrix: [0, 1, 1, 1, 1, 0, 0]\n"
     ]
    }
   ],
   "source": [
    "print(f\"encoded message by python: {HammingCode.encode(A)}\")\n",
    "print(f\"encoded message by matrix: {X}\")"
   ]
  }
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