Proposition 42.
At step \(i\) of Euclid alg.:
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\(\displaystyle s_{i}(x) a(x) + t_{i}(x) b(x) = r_{i}(x)\)
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\(\displaystyle \deg(t_{i}) + \deg(r_{i-1}) = \deg(a)\)
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\(\displaystyle \gcd(s_{i}(x), t_{i}(x)) = 1\)
Section Week 12
This is an outline of the topics we covered in the twelfth week of class.
Subsection Tuesday 3/31
Upcoming.
Today: How to do GRS decoding
Thursday: Why this worked...
ELP.
\begin{equation*}
\Lambda(x) = \prod_{j \in J}(1 - \alpha_{j} x)
\end{equation*}
EVP.
\begin{equation*}
\Gamma(x) = \sum_{j \in J}e_{j} v_{j} \prod_{m \in J \setminus \{j\}}(1 - \alpha_{m} x)
\end{equation*}
Key equation.
\begin{equation*}
\Lambda(x) S(x) \equiv \Gamma(x) \pmod{x^{d-1}}
\end{equation*}
Extended Euclid Alg:.
\begin{equation*}
q_{i}(x), \ r_{i}(x), \ s_{i}(x), \ t_{i}(x)
\end{equation*}
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\(\displaystyle s_{i}(x) a(x) + t_{i}(x) b(x) = r_{i}(x)\)
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\(\displaystyle s_{i}(x) x^{d-1}+ t_{i}(x) S(x) = r_{i}(x)\)
Final values:.
\begin{align*}
h&= 2\\
t_{h}&= 8x^{2} + 8 \implies \Lambda = x^{2} + 1 \quad (\text{using }t_{h}(0))\\
r_{h}&= 10x + 6 \implies \Gamma = 11x + 4
\end{align*}
Subsection Thursday 4/2
Coding Theorist of the Day.
Elwyn Berlekamp, 1940-2019
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PhD MIT, professor at Berkeley, Bell Labs
-
Many algorithm innovations (codes, polynomial division, …)
History / Timeline for RS Codes.
RS codes invented in 1960; some codes could be decoded with Gorenstein-Ziegler BCH decoder in 1963. The Berlekamp-Massey algorithm with complexity \(O(n^2)\) invented 1969. Sugiyama’s algorithm with the same complexity invented 1975. RS codes were used for the Voyager missions in 1977. In 1986 Berlekamp-Welch was invented with complexity \(O(n^3)\text{.}\) This was the first fast algorithm focused on the polynomial evaluation point of view. Starting in 1996 Sudan and others create list-decoding algorithms. In 2002 Gao created a fast algorithm for polynomial evaluation point of view.
Why Do They Work.
Sugiyama
Theorem 43.
For \(d \in \mathbb{Z}\text{,}\) \(S(x) \in F[x]\) there exists at most one pair of polys \(\Lambda(x), \Gamma(x)\) s.t.
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\(\displaystyle \Lambda(x) S(x) \equiv \Gamma(x) \pmod{x^{d-1}}\)
Remark 44.
Proposition 45.
Properties of Euclid alg. w/ \(a(x) = x^{d-1}\text{,}\) \(b(x) = S(x)\) give exactly 1), 2), 3) with \(\Lambda(x) = t_{h}(x) / t_{h}(0)\) and \(\Gamma(x) = r_{h}(x) / t_{h}(0)\text{.}\)
Example 46. Worksheet Problem 1 Setup.
\(y_{i} E(\alpha_{i}) = Q(\alpha_{i})\)
\(E(x) = x^{2} + e_{1} x + e_{0}\)
\(Q(x) = q_{0} + q_{1} x + q_{2} x^{2} + q_{3} x^{3} + q_{4} x^{4} + q_{5} x^{5}\)
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\(i=1\text{:}\)\begin{equation*} 0 = q_{0} + q_{1} + q_{2} + q_{3} + q_{4} + q_{5} \end{equation*}
-
\(i=2\text{:}\)\begin{equation*} 0 = q_{0} + 4q_{1} + 3q_{2} + 12q_{3} + 9q_{4} + 10q_{5} = Q(4) = Q(\alpha_{2}) \end{equation*}
-
\(i=7\text{:}\) \(y_{7} E(\alpha_{7}) = Q(\alpha_{7})\)\begin{equation*} 3 E(5) = Q(5) \end{equation*}\begin{equation*} 3(12 + 5e_{1} + e_{0}) = q_{0} + 5q_{1} + 12q_{2} + 8q_{3} + 9q_{4} + 5q_{5} \end{equation*}
\begin{equation*}
\begin{bmatrix}1&1&1&1&1&1&0&0 \\&&&&&&&\\&&&A&&&&\\&&&&&&&\\ 1&5&12&8&9&5&10&11\end{bmatrix} \begin{pmatrix}q_{0} \\ q_{1} \\ q_{2} \\ q_{3} \\ q_{4} \\ q_{5} \\ e_{0} \\ e_{1}\end{pmatrix} = \begin{pmatrix}0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 10 \\ 1\end{pmatrix}
\end{equation*}
Worksheet Problem 2 Results
\begin{align*}
Q(x)&= 8x + 4x^{2} + 5x^{3} + 10x^{4} + x^{5}\\
E(x)&= 8 + 9x + x^{2}
\end{align*}
