Claim 25. Closest Codeword in Standard Array.
The codeword at the top of a column of a standard array is no further from each received word in its column than any other code word.
Section Week 4
This is an outline of the topics we covered in the fourth week of class.
Subsection Tuesday 2/3
Coding Theorist of the Day.
Venkatesan Guruswami, PhD MIT 2001
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Rudra was his PhD student
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Key work in list-decoding
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Now at Berkeley, teaching a course on computer assisted proof this semester
Reminders/Announcements.
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Video Project - see Video Project
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Exam 1: Next T, 2/10
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4 “problems”
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Either directly from WP 1-4 or PS M1-M2 or substantially similar with number changes.
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If you turn in any M2 problems by 2 pm Thursday, will give feedback by Monday afternoon. Will do same for WP 4, but prioritize M2 if I don’t have time for both.
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Rearrange topics: swap finite fields and bounds on codes. Bounds on codes will be the next two Thursdays and finite fields after that.
Cosets.
For example, take \(W=\langle(0,1,0)\rangle_{\F_{3}}=\{000,010,020\}\)
\begin{align*}
(1,1,1)+W \amp=\{(1,1,1)+w \mid w \in W\} \\
\amp=\{111,121,101\}
\end{align*}
Is \((1,2,1)+W=(1,1,1)+W\) ? Yes,
\begin{align*}
(1,2,1)+W \amp = \{121+000, 121+010, 121+020\}\\
\amp = \{121, 101, 111\}
\end{align*}
Work on Worksheet.
Standard array from worksheet:
\(\begin{array}{c|ccc}00000 & 10110 & 01011 & 11101 \\ 00001 & 10111 & 01010 & 11100 \\ 00010 & 10100 & 01001 & 11111 \\ 00100 & 10010 & 01111 & 11001 \\ 01000 & 11110 & 00011 & 10101 \\ 10000 & 00110 & 11011 & 01101 \\ 11000 & 01110 & 10011 & 00101 \\ 10001 & 00111 & 11010 & 01100\end{array}\)
Not unique: for example, row 7 could have 00101 as the coset leader
Explanation.
Let the coset leader in row \(i\) be \(\ell_i\) and the codeword at the top of column \(j\) be \(c_j\text{,}\) so that the entry in row \(i\) and column \(j\) of the standard array is \(\ell_{i}+c_{j}\text{.}\)
\begin{align*}
d\left(\ell_{i}+c_{j}, c_{j}\right) \amp \leq d\left(\ell_{i}+c_j, c_{k}\right) \\
\wt\left(\ell_{i}\right) \amp \leq \wt\left(\ell_{i}+c_{j}-c_{k}\right)
\end{align*}
Because \(\ell_{i}\) and \(\ell_{i}+c_{j}-\ell_{k}\) are in same coset a nd we chose \(\ell_i\) to be min weight.
Claim 26. Standard Array Unique Decoding Condition.
When \(\wt\left(l_{i}\right) \leq\left\lfloor\frac{d-1}{2}\right\rfloor, l_{i}\) is unique min weight codeword in its coset, and the only codeword of weight below this threshold and we get unique decoding
Syndromes & Cosets.
Recall the syndrome of a received vector is \(\bs=H \transpose{\by}\)
What happens if \(\by_{1}, \by_{2}\) are in the same coset of \(C\) ? Then \(\by_{1}=\by_{2}+\bc\) and
\begin{align*}
\bs_{1}\amp =H \transpose{\by_{1}}\\
\amp=H\left(\transpose{\by_{2}}+\transpose{\bc}\right)\\
\amp=H \transpose{\by_{2}}+H \transpose{\bc} \\
\amp=H \transpose{\by_{2}} +0\\
\amp=\bs_{2}
\end{align*}
So we can simplify standard array:
| coset leader | syndromes |
|---|---|
| 00000 | 000 |
| 00001 | 001 |
| 00010 | 010 |
| 00100 | 100 |
| 01000 | 011 |
| 10000 | 110 |
| 11000 | 101 |
| 10001 | 111 |
\begin{equation*}
\begin{aligned}&H: 3 \times 5 \\&{\left[\begin{array}{lllll}1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1\end{array}\right]}\\&\by=11010 \\&s=111 \\&\text{ retum }11010-10001 \\&=01011\end{aligned}
\end{equation*}
Algorithm 28. Syndrome Decoding.
Precompute syndrome correspondence. Receive \(\by\text{.}\)
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Compute the received syndrome \(\bs=H \transpose{\by}\)
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Return \(\by - \be\)
Subsection Thursday 2/5
Coding Theorist of the Day.
Madhu Sudan, PhD 1992 Berkeley
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Worked in both industry & academia (IBM,MIT, Microsoft, Harvard)
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Probabilistic checking of proofs
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List-decoding algorithms
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Efficient estimation of properties of big data
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Currently teaching Data Structures & Algorithms at Harvard
Updates.
Today.
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\(\mathcal{A}_{q}(n, d)=\) max size of a code with parameters \((n, M, d)_{q}\) or \([n, k, d]_{q}\) (if linear)
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\(\displaystyle \mathcal{A}_{q}(u, d) \leq \frac{q^{n}}{V_{q}(n, \lfloor (d-1) / 2 \rfloor)}\)
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Next time: give a lower bound for \(\mathcal{A}_{q}(n, d)\) (Gilbert-Varshamov)
Perfect Codes.
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Hamming codes are perfect
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\(\displaystyle \F_q^n:[n, n, 1]_q\)
Example 29. Golay Codes.
There are two Golay codes: \([23,12,7]_{2}\) and \([11,6,5]_{3}\text{.}\) See Golay’s (short) paper introducing them. The binary code was used in the Voyager missions and secure radio and has connections to the icosahedron, sporadic groups, and quadratic residues mod 23. The ternary code has applications in quantum computing error correction.
Solution to sphere-packing equality with \(n=90, k=11, d=3\) but no code exists by symmetry argument. These codes are generated by an \((r+1)\times n\)matrix
\begin{equation*}
G=\begin{pmatrix}
g_{0} \amp g_{1} \amp \ldots \amp g_{r} \amp 0 \amp \ldots \amp 0\\
0 \amp g_{0} \amp g_{1} \amp \ldots \amp g_{r} \amp \ldots \amp 0 \\
\vdots \amp 0 \amp \ddots \amp \ddots \amp \ddots \amp \ddots \amp \vdots\\
0 \amp \ldots \amp 0 \amp g_{0} \amp g_1\amp \ldots \amp g_{r}
\end{pmatrix}
\end{equation*}
The vectors making up the nonzero entries of the row are
\begin{equation*}
(g_{0} g_{1} \ldots, g_{11})=(1101011100011)
\end{equation*}
for the \([23,12,7]_2\) code and
\begin{equation*}
(g_{0} g_{1} \ldots g_{5})=(201211)
\end{equation*}
for the \([11,6,5]_3\) code.
Theorem 30. Classification of Perfect Codes.
Proof.
This is due to Tietäväinen (1973) and van Lint (1973). See the former’s paper for details.
Singleton Bound and MDS Codes.
Theorem 31. Singleton Bound.
For an \((n, M, d)_{q}\) code \(C\)
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\(\displaystyle M \leq q^{n-d+1}\)
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\(\displaystyle d \leq n-\log _{q}(M)+1 \)
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\(d \leq n-k+1\) if the code is linear
Definition 32. Maximum Distance Separable (MDS) Codes.
Codes that meet the Singleton bound are called maximum distance separable (MDS) codes.
Examples of MDS codes include \(\mathbb{F}_{q}^{n}\text{,}\) the repetition code of length \(n\text{,}\) the parity code \(\left(x_{1}, \ldots, x_{n-1}, x_{n}\right)\) where \(x_n\) is chosen so that \(\sum_{i=0}^{n} x_{i}=0\) which is a \([n, n-1,2]_{q}\) code, and Reed-Solomon codes (Module 5 of the course).
