Reed-Solomon Berlekamp bit-serial encoder.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "3dfb2e4c",
   "metadata": {},
   "source": [
    "# Reed-Solomon Berlekamp bit-serial encoder\n",
    "\n",
    "This is a Sage notebook that calculates the quantities needed in the bit-serial encoder (Berlekamp architecture) encoder for the CCSDS Reed-Solomon code. We use the notation of the report [Perlman, M., Lee J. J., The Reed-Solomon encoders: Conventional versus Berlekamp's architecture](https://ntrs.nasa.gov/citations/19830008870). The motivation of this notebook is to correct an error in the formulas for $T_3$ and $T_4$ in this report."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c394afe3",
   "metadata": {},
   "source": [
    "CCSDS Reed-Solomon code defining parameters:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "c11c0eaf",
   "metadata": {},
   "outputs": [],
   "source": [
    "k.<alpha> = GF(256, modulus=x^8+x^7+x^2+x+1)\n",
    "K.<x> = k[]\n",
    "beta = alpha^117\n",
    "gamma = alpha^11\n",
    "g = product([x-gamma^j for j in range(112, 143+1)])  # code generator polynomial"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff9baae6",
   "metadata": {},
   "source": [
    "Coefficients of $g(x)$ as powers of $\\alpha$.\n",
    "$$g(x) = G_0 + G_1 x + G_2 x^2 + \\cdots + G_{32} x^{32}.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "46e60cdd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{split}G_{0} &= G_{32} = \\alpha^{0}\\\\G_{1} &= G_{31} = \\alpha^{249}\\\\G_{2} &= G_{30} = \\alpha^{59}\\\\G_{3} &= G_{29} = \\alpha^{66}\\\\G_{4} &= G_{28} = \\alpha^{4}\\\\G_{5} &= G_{27} = \\alpha^{43}\\\\G_{6} &= G_{26} = \\alpha^{126}\\\\G_{7} &= G_{25} = \\alpha^{251}\\\\G_{8} &= G_{24} = \\alpha^{97}\\\\G_{9} &= G_{23} = \\alpha^{30}\\\\G_{10} &= G_{22} = \\alpha^{3}\\\\G_{11} &= G_{21} = \\alpha^{213}\\\\G_{12} &= G_{20} = \\alpha^{50}\\\\G_{13} &= G_{19} = \\alpha^{66}\\\\G_{14} &= G_{18} = \\alpha^{170}\\\\G_{15} &= G_{17} = \\alpha^{5}\\\\G_{16} &= G_{16} = \\alpha^{24}\\end{split}\\]</html>"
      ],
      "text/latex": [
       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{split}G_{0} &= G_{32} = \\alpha^{0}\\\\G_{1} &= G_{31} = \\alpha^{249}\\\\G_{2} &= G_{30} = \\alpha^{59}\\\\G_{3} &= G_{29} = \\alpha^{66}\\\\G_{4} &= G_{28} = \\alpha^{4}\\\\G_{5} &= G_{27} = \\alpha^{43}\\\\G_{6} &= G_{26} = \\alpha^{126}\\\\G_{7} &= G_{25} = \\alpha^{251}\\\\G_{8} &= G_{24} = \\alpha^{97}\\\\G_{9} &= G_{23} = \\alpha^{30}\\\\G_{10} &= G_{22} = \\alpha^{3}\\\\G_{11} &= G_{21} = \\alpha^{213}\\\\G_{12} &= G_{20} = \\alpha^{50}\\\\G_{13} &= G_{19} = \\alpha^{66}\\\\G_{14} &= G_{18} = \\alpha^{170}\\\\G_{15} &= G_{17} = \\alpha^{5}\\\\G_{16} &= G_{16} = \\alpha^{24}\\end{split}$$"
      ],
      "text/plain": [
       "\\begin{split}G_{0} &= G_{32} = \\alpha^{0}\\\\G_{1} &= G_{31} = \\alpha^{249}\\\\G_{2} &= G_{30} = \\alpha^{59}\\\\G_{3} &= G_{29} = \\alpha^{66}\\\\G_{4} &= G_{28} = \\alpha^{4}\\\\G_{5} &= G_{27} = \\alpha^{43}\\\\G_{6} &= G_{26} = \\alpha^{126}\\\\G_{7} &= G_{25} = \\alpha^{251}\\\\G_{8} &= G_{24} = \\alpha^{97}\\\\G_{9} &= G_{23} = \\alpha^{30}\\\\G_{10} &= G_{22} = \\alpha^{3}\\\\G_{11} &= G_{21} = \\alpha^{213}\\\\G_{12} &= G_{20} = \\alpha^{50}\\\\G_{13} &= G_{19} = \\alpha^{66}\\\\G_{14} &= G_{18} = \\alpha^{170}\\\\G_{15} &= G_{17} = \\alpha^{5}\\\\G_{16} &= G_{16} = \\alpha^{24}\\end{split}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G_powers = [G.log(alpha) for G in list(g)]\n",
    "pretty_print(LatexExpr('\\\\begin{split}'\n",
    "                       + '\\\\\\\\'.join([f'G_{{{j}}} &= G_{{{32-j}}} = \\\\alpha^{{{n}}}'\n",
    "                                      for j, n in enumerate(G_powers[:17])])\n",
    "                       + '\\\\end{split}'))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "57397ede",
   "metadata": {},
   "source": [
    "Calculation of the basis $\\{\\ell_0, \\ell_1, \\ldots, \\ell_7\\}$ dual to $\\{\\beta^0, \\beta^1, \\ldots, \\beta^7\\}$. We express them as powers of $\\alpha$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "72fbe434",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\ell_{0} = \\alpha^{125},\\quad\\ell_{1} = \\alpha^{88},\\quad\\ell_{2} = \\alpha^{226},\\quad\\ell_{3} = \\alpha^{163},\\quad\\ell_{4} = \\alpha^{46},\\quad\\ell_{5} = \\alpha^{184},\\quad\\ell_{6} = \\alpha^{67},\\quad\\ell_{7} = \\alpha^{242}\\]</html>"
      ],
      "text/latex": [
       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\ell_{0} = \\alpha^{125},\\quad\\ell_{1} = \\alpha^{88},\\quad\\ell_{2} = \\alpha^{226},\\quad\\ell_{3} = \\alpha^{163},\\quad\\ell_{4} = \\alpha^{46},\\quad\\ell_{5} = \\alpha^{184},\\quad\\ell_{6} = \\alpha^{67},\\quad\\ell_{7} = \\alpha^{242}$$"
      ],
      "text/plain": [
       "\\ell_{0} = \\alpha^{125},\\quad\\ell_{1} = \\alpha^{88},\\quad\\ell_{2} = \\alpha^{226},\\quad\\ell_{3} = \\alpha^{163},\\quad\\ell_{4} = \\alpha^{46},\\quad\\ell_{5} = \\alpha^{184},\\quad\\ell_{6} = \\alpha^{67},\\quad\\ell_{7} = \\alpha^{242}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ells = k.dual_basis([beta^j for j in range(8)])\n",
    "ell_powers = [ell.log(alpha) for ell in ells]\n",
    "pretty_print(LatexExpr(',\\\\quad'.join([f'\\\\ell_{{{j}}} = \\\\alpha^{{{n}}}'\n",
    "                                      for j, n in enumerate(ell_powers)])))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c1836b7d",
   "metadata": {},
   "source": [
    "Calculation of the expressions $T_\\ell$ of the linear functionals $T_\\ell(z) = \\operatorname{Tr}(zG_\\ell)$, where $z = z_0 \\ell_0 + z_1 \\ell_1 + \\cdots + z_7 \\ell_7$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b4cc02c2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{split}T_{0} &=z_{0}\\\\T_{1} &=z_{1}+z_{2}+z_{4}+z_{6}\\\\T_{2} &=z_{2}+z_{3}\\\\T_{3} &=z_{0}+z_{2}+z_{4}+z_{5}\\\\T_{4} &=z_{0}+z_{2}+z_{3}+z_{7}\\\\T_{5} &=z_{0}+z_{1}+z_{2}+z_{6}+z_{7}\\\\T_{6} &=z_{0}+z_{1}+z_{5}+z_{6}\\\\T_{7} &=z_{1}+z_{2}+z_{4}\\\\T_{8} &=z_{0}+z_{1}+z_{3}+z_{6}+z_{7}\\\\T_{9} &=z_{0}+z_{2}+z_{3}+z_{4}+z_{5}\\\\T_{10} &=z_{0}+z_{1}+z_{4}+z_{7}\\\\T_{11} &=z_{4}\\\\T_{12} &=z_{0}+z_{1}+z_{2}+z_{3}+z_{4}+z_{5}+z_{6}+z_{7}\\\\T_{13} &=z_{0}+z_{2}+z_{4}+z_{5}\\\\T_{14} &=z_{0}+z_{1}+z_{2}+z_{4}+z_{5}+z_{6}\\\\T_{15} &=z_{0}+z_{1}+z_{3}+z_{5}+z_{7}\\\\T_{16} &=z_{1}+z_{2}+z_{6}\\end{split}\\]</html>"
      ],
      "text/latex": [
       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{split}T_{0} &=z_{0}\\\\T_{1} &=z_{1}+z_{2}+z_{4}+z_{6}\\\\T_{2} &=z_{2}+z_{3}\\\\T_{3} &=z_{0}+z_{2}+z_{4}+z_{5}\\\\T_{4} &=z_{0}+z_{2}+z_{3}+z_{7}\\\\T_{5} &=z_{0}+z_{1}+z_{2}+z_{6}+z_{7}\\\\T_{6} &=z_{0}+z_{1}+z_{5}+z_{6}\\\\T_{7} &=z_{1}+z_{2}+z_{4}\\\\T_{8} &=z_{0}+z_{1}+z_{3}+z_{6}+z_{7}\\\\T_{9} &=z_{0}+z_{2}+z_{3}+z_{4}+z_{5}\\\\T_{10} &=z_{0}+z_{1}+z_{4}+z_{7}\\\\T_{11} &=z_{4}\\\\T_{12} &=z_{0}+z_{1}+z_{2}+z_{3}+z_{4}+z_{5}+z_{6}+z_{7}\\\\T_{13} &=z_{0}+z_{2}+z_{4}+z_{5}\\\\T_{14} &=z_{0}+z_{1}+z_{2}+z_{4}+z_{5}+z_{6}\\\\T_{15} &=z_{0}+z_{1}+z_{3}+z_{5}+z_{7}\\\\T_{16} &=z_{1}+z_{2}+z_{6}\\end{split}$$"
      ],
      "text/plain": [
       "\\begin{split}T_{0} &=z_{0}\\\\T_{1} &=z_{1}+z_{2}+z_{4}+z_{6}\\\\T_{2} &=z_{2}+z_{3}\\\\T_{3} &=z_{0}+z_{2}+z_{4}+z_{5}\\\\T_{4} &=z_{0}+z_{2}+z_{3}+z_{7}\\\\T_{5} &=z_{0}+z_{1}+z_{2}+z_{6}+z_{7}\\\\T_{6} &=z_{0}+z_{1}+z_{5}+z_{6}\\\\T_{7} &=z_{1}+z_{2}+z_{4}\\\\T_{8} &=z_{0}+z_{1}+z_{3}+z_{6}+z_{7}\\\\T_{9} &=z_{0}+z_{2}+z_{3}+z_{4}+z_{5}\\\\T_{10} &=z_{0}+z_{1}+z_{4}+z_{7}\\\\T_{11} &=z_{4}\\\\T_{12} &=z_{0}+z_{1}+z_{2}+z_{3}+z_{4}+z_{5}+z_{6}+z_{7}\\\\T_{13} &=z_{0}+z_{2}+z_{4}+z_{5}\\\\T_{14} &=z_{0}+z_{1}+z_{2}+z_{4}+z_{5}+z_{6}\\\\T_{15} &=z_{0}+z_{1}+z_{3}+z_{5}+z_{7}\\\\T_{16} &=z_{1}+z_{2}+z_{6}\\end{split}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "T_data = [[(ell*G).trace() for ell in ells] for G in list(g)]\n",
    "pretty_print(LatexExpr(\n",
    "    '\\\\begin{split}'\n",
    "    + '\\\\\\\\'.join([f'T_{{{j}}} &=' + '+'.join([f'z_{{{k}}}' for k, a in enumerate(T)\n",
    "                                              if a == 1])\n",
    "                   for j, T in enumerate(T_data[:17])])\n",
    "    + '\\\\end{split}'))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "596eaa73",
   "metadata": {},
   "source": [
    "Calculation of the expression $z_8$ of the linear functional $z_8 = \\operatorname{Tr}(z\\beta^8)$, where $z = z_0 \\ell_0 + z_1 \\ell_1 + \\cdots + z_7 \\ell_7$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "72108ac3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}z_8 =z_{0}+z_{1}+z_{3}+z_{7}\\]</html>"
      ],
      "text/latex": [
       "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}z_8 =z_{0}+z_{1}+z_{3}+z_{7}$$"
      ],
      "text/plain": [
       "z_8 =z_{0}+z_{1}+z_{3}+z_{7}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "z8_data = [(ell*beta^8).trace() for ell in ells]\n",
    "pretty_print(LatexExpr(f'z_8 ='\n",
    "                       + '+'.join([f'z_{{{k}}}' for k, a in enumerate(z8_data)\n",
    "                                   if a == 1])))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "04db7053",
   "metadata": {},
   "source": [
    "Compute and print a table mimicking Table 4 (page 42)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "ede74903",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div class=\"truncate\"><table  class=\"table_form\">\n",
       "<tbody>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(n\\) of \\(\\alpha^n\\)</td>\n",
       "<td style=\"text-align:left\">\\(i\\) of \\(\\alpha^i\\)</td>\n",
       "<td style=\"text-align:left\">\\(\\operatorname{Tr}(\\alpha^n)\\)</td>\n",
       "<td style=\"text-align:left\">\\(j\\) of \\(\\ell_j\\)</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\"></td>\n",
       "<td style=\"text-align:left\">76543210</td>\n",
       "<td style=\"text-align:left\"></td>\n",
       "<td style=\"text-align:left\">01234567</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">00000001</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">01111011</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">00000010</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">10101111</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(2\\)</td>\n",
       "<td style=\"text-align:left\">00000100</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">10011001</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(3\\)</td>\n",
       "<td style=\"text-align:left\">00001000</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">11111010</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(4\\)</td>\n",
       "<td style=\"text-align:left\">00010000</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">10000110</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(5\\)</td>\n",
       "<td style=\"text-align:left\">00100000</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">11101100</td>\n",
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       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(6\\)</td>\n",
       "<td style=\"text-align:left\">01000000</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">11101111</td>\n",
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       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(7\\)</td>\n",
       "<td style=\"text-align:left\">10000000</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">10001101</td>\n",
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       "<tr class =\"row-a\">\n",
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       "<td style=\"text-align:left\">10000111</td>\n",
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       "<td style=\"text-align:left\">11000000</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(9\\)</td>\n",
       "<td style=\"text-align:left\">10001001</td>\n",
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       "<td style=\"text-align:left\">00001100</td>\n",
       "</tr>\n",
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       "<td style=\"text-align:left\">\\(10\\)</td>\n",
       "<td style=\"text-align:left\">10010101</td>\n",
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       "<td style=\"text-align:left\">11101001</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(11\\)</td>\n",
       "<td style=\"text-align:left\">10101101</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">01111001</td>\n",
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       "<td style=\"text-align:left\">11011101</td>\n",
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       "<td style=\"text-align:left\">11111100</td>\n",
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       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(13\\)</td>\n",
       "<td style=\"text-align:left\">00111101</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">01110010</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(14\\)</td>\n",
       "<td style=\"text-align:left\">01111010</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">11010000</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(15\\)</td>\n",
       "<td style=\"text-align:left\">11110100</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">10010001</td>\n",
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       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(16\\)</td>\n",
       "<td style=\"text-align:left\">01101111</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">10110100</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(17\\)</td>\n",
       "<td style=\"text-align:left\">11011110</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">00101000</td>\n",
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       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(18\\)</td>\n",
       "<td style=\"text-align:left\">00111011</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">01000100</td>\n",
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       "<td style=\"text-align:left\">01110110</td>\n",
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       "<td style=\"text-align:left\">10110011</td>\n",
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       "<td style=\"text-align:left\">\\(20\\)</td>\n",
       "<td style=\"text-align:left\">11101100</td>\n",
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       "<td style=\"text-align:left\">11101101</td>\n",
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       "<td style=\"text-align:left\">11011110</td>\n",
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       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(22\\)</td>\n",
       "<td style=\"text-align:left\">10111110</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">00101011</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(23\\)</td>\n",
       "<td style=\"text-align:left\">11111011</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">00100110</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(24\\)</td>\n",
       "<td style=\"text-align:left\">01110001</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">11111110</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(25\\)</td>\n",
       "<td style=\"text-align:left\">11100010</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">00100001</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(26\\)</td>\n",
       "<td style=\"text-align:left\">01000011</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">00111011</td>\n",
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       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">11011001</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(234\\)</td>\n",
       "<td style=\"text-align:left\">10001100</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">11101110</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(235\\)</td>\n",
       "<td style=\"text-align:left\">10011111</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">10111100</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(236\\)</td>\n",
       "<td style=\"text-align:left\">10111001</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">01100110</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(237\\)</td>\n",
       "<td style=\"text-align:left\">11110101</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">11101010</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(238\\)</td>\n",
       "<td style=\"text-align:left\">01101101</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">00011011</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(239\\)</td>\n",
       "<td style=\"text-align:left\">11011010</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">10110001</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(240\\)</td>\n",
       "<td style=\"text-align:left\">00110011</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">10111110</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(241\\)</td>\n",
       "<td style=\"text-align:left\">01100110</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">00110101</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(242\\)</td>\n",
       "<td style=\"text-align:left\">11001100</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">00000001</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(243\\)</td>\n",
       "<td style=\"text-align:left\">00011111</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">00110001</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(244\\)</td>\n",
       "<td style=\"text-align:left\">00111110</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">10100110</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(245\\)</td>\n",
       "<td style=\"text-align:left\">01111100</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">11100110</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(246\\)</td>\n",
       "<td style=\"text-align:left\">11111000</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">11110010</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(247\\)</td>\n",
       "<td style=\"text-align:left\">01110111</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">11001000</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(248\\)</td>\n",
       "<td style=\"text-align:left\">11101110</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">01000010</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(249\\)</td>\n",
       "<td style=\"text-align:left\">01011011</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">01000111</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(250\\)</td>\n",
       "<td style=\"text-align:left\">10110110</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">11010001</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(251\\)</td>\n",
       "<td style=\"text-align:left\">11101011</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">10100000</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(252\\)</td>\n",
       "<td style=\"text-align:left\">01010001</td>\n",
       "<td style=\"text-align:left\">\\(0\\)</td>\n",
       "<td style=\"text-align:left\">00010010</td>\n",
       "</tr>\n",
       "<tr class =\"row-b\">\n",
       "<td style=\"text-align:left\">\\(253\\)</td>\n",
       "<td style=\"text-align:left\">10100010</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">11001110</td>\n",
       "</tr>\n",
       "<tr class =\"row-a\">\n",
       "<td style=\"text-align:left\">\\(254\\)</td>\n",
       "<td style=\"text-align:left\">11000011</td>\n",
       "<td style=\"text-align:left\">\\(1\\)</td>\n",
       "<td style=\"text-align:left\">10110110</td>\n",
       "</tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "  $n$ of $\\alpha^n$   $i$ of $\\alpha^i$   $\\operatorname{Tr}(\\alpha^n)$   $j$ of $\\ell_j$\n",
       "                      76543210                                            01234567\n",
       "  0                   00000001            0                               01111011\n",
       "  1                   00000010            1                               10101111\n",
       "  2                   00000100            1                               10011001\n",
       "  3                   00001000            1                               11111010\n",
       "  4                   00010000            1                               10000110\n",
       "  5                   00100000            1                               11101100\n",
       "  6                   01000000            1                               11101111\n",
       "  7                   10000000            1                               10001101\n",
       "  8                   10000111            1                               11000000\n",
       "  9                   10001001            0                               00001100\n",
       "  10                  10010101            1                               11101001\n",
       "  11                  10101101            0                               01111001\n",
       "  12                  11011101            1                               11111100\n",
       "  13                  00111101            0                               01110010\n",
       "  14                  01111010            1                               11010000\n",
       "  15                  11110100            1                               10010001\n",
       "  16                  01101111            1                               10110100\n",
       "  17                  11011110            0                               00101000\n",
       "  18                  00111011            0                               01000100\n",
       "  19                  01110110            1                               10110011\n",
       "  20                  11101100            1                               11101101\n",
       "  21                  01011111            1                               11011110\n",
       "  22                  10111110            0                               00101011\n",
       "  23                  11111011            0                               00100110\n",
       "  24                  01110001            1                               11111110\n",
       "  25                  11100010            0                               00100001\n",
       "  26                  01000011            0                               00111011\n",
       "  27                  10000110            1                               10111011\n",
       "  28                  10001011            1                               10100011\n",
       "  29                  10010001            0                               01110000\n",
       "  30                  10100101            1                               10000011\n",
       "  31                  11001101            0                               01111010\n",
       "  32                  00011101            1                               10011110\n",
       "  33                  00111010            0                               00111111\n",
       "  34                  01110100            0                               00011100\n",
       "  35                  11101000            0                               01110100\n",
       "  36                  01010111            0                               00100100\n",
       "  37                  10101110            1                               10101101\n",
       "  38                  11011011            1                               11001010\n",
       "  39                  00110001            0                               00010001\n",
       "  40                  01100010            1                               10101100\n",
       "  41                  11000100            1                               11111011\n",
       "  42                  00001111            1                               10110111\n",
       "  43                  00011110            0                               01001010\n",
       "  44                  00111100            0                               00001001\n",
       "  45                  01111000            0                               01111111\n",
       "  46                  11110000            0                               00001000\n",
       "  47                  01100111            0                               01001110\n",
       "  48                  11001110            1                               10101110\n",
       "  49                  00011011            1                               10101000\n",
       "  50                  00110110            0                               01011100\n",
       "  51                  01101100            0                               01100000\n",
       "  52                  11011000            0                               00011110\n",
       "  53                  00110111            0                               00100111\n",
       "  54                  01101110            1                               11001111\n",
       "  55                  11011100            1                               10000111\n",
       "  56                  00111111            1                               11011101\n",
       "  57                  01111110            0                               01001001\n",
       "  58                  11111100            0                               01101011\n",
       "  59                  01111111            0                               00110010\n",
       "  60                  11111110            1                               11000100\n",
       "  61                  01111011            1                               10101011\n",
       "  62                  11110110            0                               00111110\n",
       "  63                  01101011            0                               00101101\n",
       "  64                  11010110            1                               11010010\n",
       "  65                  00101011            1                               11000010\n",
       "  66                  01010110            0                               01011111\n",
       "  67                  10101100            0                               00000010\n",
       "  68                  11011111            0                               01010011\n",
       "  69                  00111001            1                               11101011\n",
       "  70                  01110010            0                               00101010\n",
       "  71                  11100100            0                               00010111\n",
       "  72                  01001111            0                               01011000\n",
       "  73                  10011110            1                               11000111\n",
       "  74                  10111011            1                               11001001\n",
       "  75                  11110001            0                               01110011\n",
       "  76                  01100101            1                               11100001\n",
       "  77                  11001010            0                               00110111\n",
       "  78                  00010011            0                               01010010\n",
       "  79                  00100110            1                               11011010\n",
       "  80                  01001100            1                               10001100\n",
       "  81                  10011000            1                               11110001\n",
       "  82                  10110111            1                               10101010\n",
       "  83                  11101001            0                               00001111\n",
       "  84                  01010101            1                               10001011\n",
       "  85                  10101010            0                               00110100\n",
       "  86                  11010011            0                               00110000\n",
       "  87                  00100001            1                               10010111\n",
       "  88                  01000010            0                               01000000\n",
       "  89                  10000100            0                               00010100\n",
       "  90                  10001111            0                               00111010\n",
       "  91                  10011001            1                               10001010\n",
       "  92                  10110101            0                               00000101\n",
       "  93                  11101101            1                               10010110\n",
       "  94                  01011101            0                               01110001\n",
       "  95                  10111010            1                               10110010\n",
       "  96                  11110011            1                               11011100\n",
       "  97                  01100001            0                               01111000\n",
       "  98                  11000010            1                               11001101\n",
       "  99                  00000011            1                               11010100\n",
       "  100                 00000110            0                               00110110\n",
       "  101                 00001100            0                               01100011\n",
       "  102                 00011000            0                               01111100\n",
       "  103                 00110000            0                               01101010\n",
       "  104                 01100000            0                               00000011\n",
       "  105                 11000000            0                               01100010\n",
       "  106                 00000111            0                               01001101\n",
       "  107                 00001110            1                               11001100\n",
       "  108                 00011100            1                               11100101\n",
       "  109                 00111000            1                               10010000\n",
       "  110                 01110000            1                               10000101\n",
       "  111                 11100000            1                               10001110\n",
       "  112                 01000111            1                               10100010\n",
       "  113                 10001110            0                               01000001\n",
       "  114                 10011011            0                               00100101\n",
       "  115                 10110001            1                               10011100\n",
       "  116                 11100101            0                               01101100\n",
       "  117                 01001101            1                               11110111\n",
       "  118                 10011010            0                               01011110\n",
       "  119                 10110011            0                               00110011\n",
       "  120                 11100001            1                               11110101\n",
       "  121                 01000101            0                               00001101\n",
       "  122                 10001010            1                               11011000\n",
       "  123                 10010011            1                               11011111\n",
       "  124                 10100001            0                               00011010\n",
       "  125                 11000101            1                               10000000\n",
       "  126                 00001101            0                               00011000\n",
       "  127                 00011010            1                               11010011\n",
       "  128                 00110100            1                               11110011\n",
       "  129                 01101000            1                               11111001\n",
       "  130                 11010000            1                               11100100\n",
       "  131                 00100111            1                               10100001\n",
       "  132                 01001110            0                               00100011\n",
       "  133                 10011100            0                               01101000\n",
       "  134                 10111111            0                               01010000\n",
       "  135                 11111001            1                               10001001\n",
       "  136                 01110101            0                               01100111\n",
       "  137                 11101010            1                               11011011\n",
       "  138                 01010011            1                               10111101\n",
       "  139                 10100110            0                               01010111\n",
       "  140                 11001011            0                               01001100\n",
       "  141                 00010001            1                               11111101\n",
       "  142                 00100010            0                               01000011\n",
       "  143                 01000100            0                               01110110\n",
       "  144                 10001000            0                               01110111\n",
       "  145                 10010111            0                               01000110\n",
       "  146                 10101001            1                               11100000\n",
       "  147                 11010101            0                               00000110\n",
       "  148                 00101101            1                               11110100\n",
       "  149                 01011010            0                               00111100\n",
       "  150                 10110100            0                               01111110\n",
       "  151                 11101111            0                               00111001\n",
       "  152                 01011001            1                               11101000\n",
       "  153                 10110010            0                               01001000\n",
       "  154                 11100011            0                               01011010\n",
       "  155                 01000001            1                               10010100\n",
       "  156                 10000010            0                               00100010\n",
       "  157                 10000011            0                               01011001\n",
       "  158                 10000001            1                               11110110\n",
       "  159                 10000101            0                               01101111\n",
       "  160                 10001101            1                               10010101\n",
       "  161                 10011101            0                               00010011\n",
       "  162                 10111101            1                               11111111\n",
       "  163                 11111101            0                               00010000\n",
       "  164                 01111101            1                               10011101\n",
       "  165                 11111010            0                               01011101\n",
       "  166                 01110011            0                               01010001\n",
       "  167                 11100110            1                               10111000\n",
       "  168                 01001011            1                               11000001\n",
       "  169                 10010110            0                               00111101\n",
       "  170                 10101011            0                               01001111\n",
       "  171                 11010001            1                               10011111\n",
       "  172                 00100101            0                               00001110\n",
       "  173                 01001010            1                               10111010\n",
       "  174                 10010100            1                               10010010\n",
       "  175                 10101111            1                               11010110\n",
       "  176                 11011001            0                               01100101\n",
       "  177                 00110101            1                               10001000\n",
       "  178                 01101010            0                               01010110\n",
       "  179                 11010100            0                               01111101\n",
       "  180                 00101111            0                               01011011\n",
       "  181                 01011110            1                               10100101\n",
       "  182                 10111100            1                               10000100\n",
       "  183                 11111111            1                               10111111\n",
       "  184                 01111001            0                               00000100\n",
       "  185                 11110010            1                               10100111\n",
       "  186                 01100011            1                               11010111\n",
       "  187                 11000110            0                               01010100\n",
       "  188                 00001011            0                               00101110\n",
       "  189                 00010110            1                               10110000\n",
       "  190                 00101100            1                               10001111\n",
       "  191                 01011000            1                               10010011\n",
       "  192                 10110000            1                               11100111\n",
       "  193                 11100111            1                               11000011\n",
       "  194                 01001001            0                               01101110\n",
       "  195                 10010010            1                               10100100\n",
       "  196                 10100011            1                               10110101\n",
       "  197                 11000001            0                               00011001\n",
       "  198                 00000101            1                               11100010\n",
       "  199                 00001010            0                               01010101\n",
       "  200                 00010100            0                               00011111\n",
       "  201                 00101000            0                               00010110\n",
       "  202                 01010000            0                               01101001\n",
       "  203                 10100000            0                               01100001\n",
       "  204                 11000111            0                               00101111\n",
       "  205                 00001001            1                               10000001\n",
       "  206                 00010010            0                               00101001\n",
       "  207                 00100100            0                               01110101\n",
       "  208                 01001000            0                               00010101\n",
       "  209                 10010000            0                               00001011\n",
       "  210                 10100111            0                               00101100\n",
       "  211                 11001001            1                               11100011\n",
       "  212                 00010101            0                               01100100\n",
       "  213                 00101010            1                               10111001\n",
       "  214                 01010100            1                               11110000\n",
       "  215                 10101000            1                               10011011\n",
       "  216                 11010111            1                               10101001\n",
       "  217                 00101001            0                               01101101\n",
       "  218                 01010010            1                               11000110\n",
       "  219                 10100100            1                               11111000\n",
       "  220                 11001111            1                               11010101\n",
       "  221                 00011001            0                               00000111\n",
       "  222                 00110010            1                               11000101\n",
       "  223                 01100100            1                               10011010\n",
       "  224                 11001000            1                               10011000\n",
       "  225                 00010111            1                               11001011\n",
       "  226                 00101110            0                               00100000\n",
       "  227                 01011100            0                               00001010\n",
       "  228                 10111000            0                               00011101\n",
       "  229                 11110111            0                               01000101\n",
       "  230                 01101001            1                               10000010\n",
       "  231                 11010010            0                               01001011\n",
       "  232                 00100011            0                               00111000\n",
       "  233                 01000110            1                               11011001\n",
       "  234                 10001100            1                               11101110\n",
       "  235                 10011111            1                               10111100\n",
       "  236                 10111001            0                               01100110\n",
       "  237                 11110101            1                               11101010\n",
       "  238                 01101101            0                               00011011\n",
       "  239                 11011010            1                               10110001\n",
       "  240                 00110011            1                               10111110\n",
       "  241                 01100110            0                               00110101\n",
       "  242                 11001100            0                               00000001\n",
       "  243                 00011111            0                               00110001\n",
       "  244                 00111110            1                               10100110\n",
       "  245                 01111100            1                               11100110\n",
       "  246                 11111000            1                               11110010\n",
       "  247                 01110111            1                               11001000\n",
       "  248                 11101110            0                               01000010\n",
       "  249                 01011011            0                               01000111\n",
       "  250                 10110110            1                               11010001\n",
       "  251                 11101011            1                               10100000\n",
       "  252                 01010001            0                               00010010\n",
       "  253                 10100010            1                               11001110\n",
       "  254                 11000011            1                               10110110"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alphai = [''.join([str(a) for a in (list((alpha^n).polynomial()) + [0]*8)[:8][::-1]]) for n in range(255)]\n",
    "trace = [(alpha^n).trace() for n in range(255)]\n",
    "ellj = [''.join([str((beta^j*alpha^n).trace()) for j in range(8)]) for n in range(255)]\n",
    "\n",
    "header = [['$n$ of $\\\\alpha^n$', '$i$ of $\\\\alpha^i$',\n",
    "           '$\\\\operatorname{Tr}(\\\\alpha^n)$', '$j$ of $\\\\ell_j$'],\n",
    "          ['', '76543210', '', '01234567']]\n",
    "body = [[n, x[0], x[1], x[2]] for n, x in enumerate(zip(alphai, trace, ellj))]\n",
    "table(header + body)"
   ]
  }
 ],
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