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@misc{reedsolomon, |
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title = {{Reed-Solomon Codes paper}}, |
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author = {I. S. Reed, G. Solomon}, |
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howpublished = "\url{https://faculty.math.illinois.edu/~duursma/CT/RS-1960.pdf}", |
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} |
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|
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@misc{reedsolomon-bruce, |
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title = {Reed-Solomon}, |
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author = {Bruce Maggs}, |
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howpublished = "\url{https://courses.cs.duke.edu/spring10/cps296.3/rs_scribe.pdf}", |
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} |
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|
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@misc{vanddetproof, |
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title = {Vandermonde determinant proof}, |
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author = {Pedro Tamaroff}, |
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howpublished = "\url{https://math.stackexchange.com/questions/525334/vandermonde-determinant-by-induction}", |
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} |
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|
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@misc{vanddetproof2, |
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title = {Abstract Algebra, Theory and Applications. \emph{sec.22.2}}, |
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author = {Thomas W. Judson}, |
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} |
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|
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@misc{cyclotomicpolyroots, |
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title = {Lagrange bases in subgroups of Fp*: a hands-on introduction}, |
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author = {Alex Kampa}, |
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howpublished = "\url{https://github.com/aragon/research/blob/main/blog/001_Lagrange/lagrange.pdf}", |
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} |
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notes_reed-solomon.pdf
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notes_reed-solomon.pdf
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\documentclass[a4paper]{article} |
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\setlength{\parindent}{0em} |
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\setlength{\parskip}{0.4em} |
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\usepackage{amsmath} |
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\usepackage{amsfonts} |
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\usepackage{amsthm} |
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\usepackage{enumerate} |
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\usepackage[bookmarksopen=true,bookmarks=true]{hyperref} |
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\usepackage{bookmark} |
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\hypersetup{ |
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pdfborder = {0.1 0.1 0.1}, |
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% colorlinks, |
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% citecolor=black, |
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% filecolor=black, |
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% linkcolor=black, |
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% urlcolor=blue |
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} |
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\usepackage{multicol} |
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\usepackage{graphicx} |
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\usepackage{float} |
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\graphicspath{ {./imgs} } |
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\usepackage{pgf-umlsd} % diagrams |
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\usepackage[UKenglish]{datetime} |
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\newdateformat{DATEMonthYear}{\monthname[\THEMONTH] \THEYEAR} |
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\newdateformat{DATEYearMonthDay}{\THEYEAR-\THEMONTH-\THEDAY} |
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\usepackage{booktabs} % \toprule \midrule ... |
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\theoremstyle{definition} |
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\newtheorem{definition}{Def}[section] |
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\newtheorem{theorem}[definition]{Thm} |
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\title{Notes on Reed-Solomon codes} |
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\author{arnaucube} |
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\date{January 2023} |
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\begin{document} |
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\maketitle |
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\begin{abstract} |
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Notes taken while reading about Reed-Solomon codes. Usually while reading papers I take handwritten notes, this document contains some of them re-written to $LaTeX$. |
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The notes are not complete, don't include all the steps neither all the proofs. |
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\end{abstract} |
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\tableofcontents |
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\section{Reed-Solomon Codes overview} |
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In this section we overview the main ideas presented in the Reed-Solomon paper \cite{reedsolomon} and in Bruce Maggs notes \cite{reedsolomon-bruce}. |
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Reed-Solomon codes appeared in parallel to the BCH codes, and can be described as nonbinary BCH codes. In this section we will focus only in the Reed-Solomon codes, particularly in the encoding and error detection (not correction). |
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\emph{tmp-note}: I feel like it is worth to check Galois theory \& BCH codes before going to Reed-Solomon codes, but due time constraints and our main goal being FRI, probably we should skip it. |
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\subsection{Idea} |
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Let $p(x) = m_0 + m_1 x + \ldots + m_{k-1} x^{k-1}$, where $m_i \in K$ and $k < 2^n$. |
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We map $k$-tuples of $K$ into $2^n$-tuples of $K$, where $K=GF(p^r)$. |
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% Code E maps $k$-tuples of $K$ into $2^n$-tuples of $K$, where $K$ is a field of degree $n$ over the field of two elements $\mathbb{Z}_2$. |
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$$ |
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k |
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\begin{cases} |
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m_0\\ |
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m_1\\ |
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\vdots\\ |
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m_{k-1} |
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\end{cases} |
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\in K~~~~ |
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% \stackrel{E}{ \xrightarrow{\hspace*{1cm}} } |
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\xrightarrow{\hspace*{1cm}} |
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2^n |
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\begin{cases} |
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p(0)\\ |
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p(\alpha)\\ |
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p(\alpha^2)\\ |
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\vdots\\ |
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p(1) |
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\end{cases} |
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\in K |
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$$ |
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The receiver then, can decode the messages by solving simultaneously any $k$ of the $2^n$ equations |
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\begin{align*} |
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p(0) &= m_0\\ |
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p(\alpha) &= m_0 + m_1 \alpha + m_2 \alpha^2 + \ldots + m_{k-1} \alpha^{k-1}\\ |
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p(\alpha^2) &= m_0 + m_1 \alpha^2 + m_2 \alpha^4 + \ldots + m_{k-1} \alpha^{2k-2}\\ |
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&~~~\vdots\\ |
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p(1) &= m_0 + m_1 + m_2 + \ldots + m_{k-1}\\ |
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\end{align*} |
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This system of equations can be solved, as we can see that any $k$ of the $p(\alpha^j)$ equations are linearly independent since they form a Vandermonde matrix |
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$$ |
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V = |
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\begin{pmatrix} |
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1 & x_1 & x_1^2 & \ldots & x_1^{k-1} \\ |
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1 & x_2 & x_2^2 & \ldots & x_2^{k-1} \\ |
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1 & x_3 & x_3^2 & \ldots & x_3^{k-1} \\ |
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\vdots & \vdots & \vdots & & \vdots\\ |
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1 & x_n & x_n^2 & \ldots & x_n^{k-1} \\ |
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\end{pmatrix} |
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$$ |
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and $\det V$ is a Vandermonde determinant and $\det V \neq 0$ as |
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$$ |
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\det~V = \prod_{j < i} (x_i - x_j) |
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$$ |
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See Annex \ref{sec:vandermondedet} for a proof of the Vandermonde determinant. |
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% The number of combinations of taking $k$ elements from $2^n$ elements (without regard of order) can be expressed by $C_k^{2^n} = \binom{2^n}{k} = \frac{(2^n)!}{k!~(2^n - k)!}$. |
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% This is the number of determinations of $(m_0, \ldots, m_{m-1})$ in case of no errors. |
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% TODO this part needs to be continued & finished. |
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\subsection{Real world approach} |
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In the practical side, instead of transmitting $k+2s$ polynomial evaluations, we send $k$ coefficients + $2s$ evaluations. |
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This is because we are interested in efficiency in the \emph{common} case, where in most of cases there are no errors and we care more about having a fast check that there are no errors than of the error recovering phase. |
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Furthermore, in our use case in the context of FRI IOP, we are not interested into decoding but only into detecting errors. |
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% \subsubsection{The setup} |
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% We work on $GF(p^r)$. For $\alpha$ being a primitive element of $GF(p^r)$, we set the generator polynomial |
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% |
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% $$ |
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% g(x) = (x-\alpha) (x-\alpha^2) \cdots (x-\alpha^{2s-1}) |
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% $$ |
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% |
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% |
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% THIS part is not coherent with the 'Encoding' section |
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% We define |
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% $$b(x) = x^{2s} \cdot m(x) \pmod{g(x)}$$ |
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% |
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% thus, for some $q(x)$, |
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% $$x^{2s} m(x) = q(x) g(x) + b(x)$$ |
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% |
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% Define the codeword $c(x)$ as |
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% $$c(x) = x^{2s} \cdot m(x) - b(x)$$ |
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% UNTIL HERE. |
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|
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% notice that $c(x)=q(x)g(x)$, thus $c(x)$ is a multiple of $g(x)$. |
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% |
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% In order to check if a received codeword $c'(x)$ is correct, we'll check its divisibility by $g(x)$. If the check passes, we extract the first $k$ elements of $c'(x)$. |
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\subsubsection{Encoding} |
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Let $g(x)$ be the generator polynomial |
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$$g(x) = (x-\alpha) (x-\alpha^2) \cdots (x-\alpha^{2s-1})$$ |
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whith $\alpha$ being a primitive element of $GF(p^r)$. |
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The \emph{encoder} wants to map the message $\{ m_0, m_1, \ldots, m_{k-1} \}$ into a polynomial $p(x)$ of degree $<k$ such that |
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$$p(\alpha_i) = m_i ~~ \forall i\in{0, k-1}$$ |
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so we interpolate taking $\{ m_0, m_1, \ldots, m_{k-1} \}$ as the coefficients of $p(x)$ in order to obtain |
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$$p(x) = m_0 + m_1 \cdot x + \cdots + m_{k-1} x^{k-1}$$ |
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Then the encoder computes $c(x) = p(x) \cdot g(x)$, and will output the coefficients of $c(x)$, $\{ c_i \}^k$. |
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Notice that $c(x)=q(x)g(x)$, thus $c(x)$ is a multiple of $g(x)$. |
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\subsubsection{Error checking} |
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The \emph{receiver} starts from the received coefficients $\{c_i'\}^k$ of $c'(x)$. $c'(x)$ may contain errors, which we represent by $e(x) = e_0 + e_1 x + e_2 x^2 + \ldots + e_{k-1} x^{k-1}$, so $c'(x) = c(x) + e(x)$. |
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In order to check $c(x)\stackrel{?}{=} c'(x)$ (thus, that there are no errors ($e(x) = 0$)), we will do the divisibility check $g(x) | c'(x)$, which if there are no errors the check should pass. This comes from the fact that we've chosen $c(x)$ to be a multiple of $g(x)$. |
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If the check passes, we'll extract the first $k$ elements of $c'(x)$. |
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If $c'(x) / g(x) = 0$, it means that $c'(x)$ is a multiple of $g(x)$, thus $c(x)=c'(x)$, which means that there are no errors. |
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In practice, checking divisibility by dividing $c'(x)$ by $g(x)$ could be computationally expensive, that's why we do the following approach. |
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% so we multiply $c(x)$ by some polynomial $d(x)$, and then we check if $g(x)d(x)$ divides the result. |
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% |
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% In this way, instead of checking that $c'(x)/g(x) = 0$, we check that |
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% $$c'(x) = d(x) \cdot g(x) = 0 \pmod{x^n - 1}$$ |
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% |
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% If $c'(x) / g(x) \neq 0$, means that there are errors, such that |
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% $$c'(x) = p(x) \cdot g(x) + e(x)$$ |
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\textbf{The check polynomial} |
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From $g(x) = (x-\alpha) (x-\alpha^2) \cdots (x-\alpha^{2s-1})$, we know that |
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$$g(x) = \prod_{i=0}^{2s-1} (x - \alpha^i) = x^{2s} - 1$$ |
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a proof for $\prod_{i=0}^{n-1} (x - \alpha^i) = x^n - 1$ can be found in \cite{cyclotomicpolyroots}. |
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Now, let $g'(x)=\prod_{i=0}^{n-1} (x-\alpha^i)$, which for $n=k +2s$, is equivalent to |
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$$g'(x) = \prod_{i=0}^{n-1} (x-\alpha^i) = \prod_{i=0}^{2s-1} (x-\alpha^i) ~\cdot \prod_{i=2s-1}^{n-1} (x-\alpha^i)$$ |
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% \begin{align*} |
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% g'(x) = \prod_{i=0}^{n-1} (x-\alpha^i) =& \prod_{i=0}^{2s-1} (x-\alpha^i) ~\cdot \prod_{i=2s-1}^{n-1} (x-\alpha^i)\\ |
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% =& ~g(x) ~\cdot \prod_{i=2s-1}^{n-1} (x-\alpha^i) |
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% \end{align*} |
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where we see clearly that $g'(x)$ is a multiple of $g(x)$, so, we have that $g(x) | g'(x)$, which is the same than saying that |
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$$g(x) | (x^n-1),~for~2s<n$$ |
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From here, as we know that $c(x)$ is a multiple of $g(x)$, we can see that there is a unique polynomial $h(x)$ of degree $n-2s$ such that |
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$$g(x) h(x) = x^n - 1$$ |
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So, $h(x) = (x^n -1) / g(x)$, and we can see that |
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$$h(x) = (x^n -1)/g(x)= \prod_{2s-1}^{n-1} (x - \alpha^i)$$ |
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% $$h(x) = (x^n -1)/g(x)=\frac{x^n -1}{x^{2s} -1} = \frac{\prod_0^{n-1} (x-\alpha^i)}{\prod_0^{2s-1} (x-\alpha^i)}$$ |
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% $$= \frac{\prod_0^{2s-1} (x-\alpha^i) \cdot \prod_{2s-1}^{n-1} (x - \alpha^i)}{\prod_0^{2s-1} (x-\alpha^i)} |
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% = \prod_{2s-1}^{n-1} (x - \alpha^i) |
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% $$ |
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The degree of $h(x)$ being $n-2s$ can be seen from the fact that $deg(g(x))=2s$ and $deg(x^n -1)=n$, thus |
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$$deg(h(x)) = deg((x^n -1)/g(x))=n-2s$$ |
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which is equal to $k$. |
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The polynomial $h(x)$ receives the \emph{check polynomial} name. |
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From here, to determine if $c'(x)$ is a valid codeword we check: |
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$$c'(x) \cdot h(x) = 0 \pmod{x^n - 1}$$ |
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It is common to choose $n = p^r -1$ such as $n=2^8 - 1$, and $k=223$ (\emph{RS(255, 223)}), over $GF(2^8)$. |
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\newpage |
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%%%%%% APPENDIX |
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\appendix |
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\section{Vandermonde determinant} |
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\label{sec:vandermondedet} |
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This section contains a simple proof of the Vandermonde matrix determinant. The proof was found in \cite{vanddetproof}, here it is written with small modifications and expanded. A different proof can be found in \cite{vanddetproof2}. |
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$$ |
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\det~V(x_1, ..., x_n) = \det |
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\begin{pmatrix} |
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1 & x_1 & x_1^2 & \ldots & x_1^{n-1} \\ |
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1 & x_2 & x_2^2 & \ldots & x_2^{n-1} \\ |
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1 & x_3 & x_3^2 & \ldots & x_3^{n-1} \\ |
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\vdots & \vdots & \vdots & & \vdots\\ |
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1 & x_n & x_n^2 & \ldots & x_n^{n-1} \\ |
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\end{pmatrix} |
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= \prod_{j < i} (x_i - x_j) |
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$$ |
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\emph{Proof} |
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The equation is true for $n = 2$, as we have: |
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$$ |
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\det~V(x_1, x_2) = \det |
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\begin{pmatrix} |
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1 & x_1\\ |
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1 & x_2 |
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\end{pmatrix} |
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= x_2 - x_1 |
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$$ |
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We now assume that $n \ge 3$. Starting with column $n$ and ending at column $2$, we successively apply the following operation: subtract $x_1$ times column $i-1$ from column $i$. The determinant remains the same and the resulting matrix is: |
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$$ |
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= \det |
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\begin{pmatrix} |
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1 & x_1-(x_1 \cdot 1) & x_1^2 - (x_1 \cdot x_1) & \ldots & x_1^{n-1} - (x_1 \cdot x_1^{n-2}) \\ |
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1 & x_2-(x_1 \cdot 1) & x_2^2 - (x_1 \cdot x_2) & \ldots & x_2^{n-1} - (x_1 \cdot x_2^{n-2}) \\ |
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1 & x_3-(x_1 \cdot 1) & x_3^2 - (x_1 \cdot x_3) & \ldots & x_3^{n-1} - (x_1 \cdot x_3^{n-2}) \\ |
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\vdots & \vdots & \vdots & & \vdots\\ |
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1 & x_n-(x_1 \cdot 1) & x_n^2 - (x_1 \cdot x_n) & \ldots & x_n^{n-1} - (x_1 \cdot x_n^{n-2}) \\ |
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\end{pmatrix} |
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$$ |
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$$ |
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= \det |
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\begin{pmatrix} |
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1 & 0 & 0 & \ldots & 0 \\ |
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1 & (x_2 - x_1) & x_2(x_2 - x_1) & \ldots & x_2^{n-2}(x_2 - x_1) \\ |
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1 & (x_3 - x_1) & x_3(x_3 - x_1) & \ldots & x_3^{n-2}(x_3 - x_1) \\ |
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\vdots & \vdots & \vdots & & \vdots \\ |
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1 & (x_n - x_1) & x_n(x_n - x_1) & \ldots & x_n^{n-2}(x_n - x_1) \\ |
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\end{pmatrix} |
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$$ |
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Applying the Laplace expansion along the first row and factoring out the scalar multiples $(x_i - x_1)$ in each column, we obtain: |
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$$ |
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= \det |
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\begin{pmatrix} |
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x_2-x_1 & x_2^2 - x_1 x_2 & x_2^3 - x_1 x_2^2 & \ldots & x_2^{n-1} - x_1 x_2^{n-2} \\ |
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x_3-x_1 & x_3^2 - x_1 x_3 & x_3^3 - x_1 x_3^2 & \ldots & x_3^{n-1} - x_1 x_3^{n-2} \\ |
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\vdots & \vdots & \vdots & & \vdots\\ |
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x_n-x_1 & x_n^2 - x_1 x_n & x_n^3 - x_1 x_n^2 & \ldots & x_n^{n-1} - x_1 x_n^{n-2} \\ |
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\end{pmatrix} |
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$$ |
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in each row, factor out $(x_i - x_1)$ |
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$$ |
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= \det |
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\begin{pmatrix} |
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1(x_2-x_1) & x_2(x_2 - x_1) & x_2^2(x_2 - x_1) & \ldots & x_2^{n-2}(x_2 - x_1) \\ |
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1(x_3-x_1) & x_3(x_3 - x_1) & x_3^2(x_3 - x_1) & \ldots & x_3^{n-2}(x_3 - x_1) \\ |
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\vdots & \vdots & \vdots & & \vdots\\ |
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1(x_n-x_1) & x_n(x_n - x_1) & x_n^2(x_n - x_1) & \ldots & x_n^{n-2}(x_n - x_1) \\ |
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\end{pmatrix} |
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$$ |
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$$ |
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= \prod_{2 \leq i \leq n} (x_i - x_1) ~~\det |
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\begin{pmatrix} |
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1 & x_2 & x_2^2 & \ldots & x_2^{n-2} \\ |
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1 & x_3 & x_3^2 & \ldots & x_3^{n-2} \\ |
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\vdots & \vdots & \vdots & & \vdots\\ |
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1 & x_n & x_n^2 & \ldots & x_n^{n-2} \\ |
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\end{pmatrix} |
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$$ |
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now, by induction |
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$$ |
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= \prod_{i=2}^n (x_i - x_1) ~~\cdot \prod_{2 \leq j < i \leq n} (x_i - x_j) ~~= \prod_{1 \leq j < i \leq n} (x_i - x_j) |
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$$ |
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\qed |
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% Bibliography |
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\addcontentsline{toc}{section}{References} |
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\bibliography{notes_reed-solomon.bib} |
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\bibliographystyle{unsrt} |
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\end{document} |
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