begin correction

correction.bib

@@ -0,0 +1,8 @@
+@misc{ enwiki:signal-to-noise,
+  author = "{Wikipedia contributors}",
+  title = "Signal-to-noise ratio --- {Wikipedia}{,} The Free Encyclopedia",
+  year = "2026",
+  url = "https://en.wikipedia.org/w/index.php?title=Signal-to-noise_ratio&oldid=1334458479",
+  note = "[Online; accessed 16-February-2026]"
+}
+

correction.tex

@@ -14,8 +14,11 @@
 \usepackage{amsmath}
 \PassOptionsToPackage{hyphens}{url}
 \usepackage{hyperref} % allows urls to follow line breaks of text
+\usepackage{glossaries}
 \usepackage[style=ieee, backend=biber, maxnames=1, minnames=1]{biblatex}
-\addbibresource{entropy.bib}
+\addbibresource{correction.bib}
+
+\newacronym{snr}{SNR}{signal-to-noise ratio}
 
 
 
@@ -25,7 +28,34 @@
 
 \begin{document}
 \maketitle
-(theorems: Hamming condition, Varsham-Gilbert)
+\section{Introduction}
+When messages are transmitted over real media, errors are inevitably introduced.
+Though the source of errors are manifold, they are ubiquitous and need to be accounted for if we want any
+real chance of reading a correct message at the end.
+On a basic level, we can understand that a channel with a lower \textit{\acrfull{snr}},
+i.e. more errors per intended information will require more effort to retain the message information.
+\[ \mathrm{SNR} = \frac{P_{signal}}{P_{noise}} \]
+In an analog system, one might attempt to simply increase transmission power $P_{signal}$
+as is done for example in professional audio equipment.
+This would however require the modification of the transmission channel itself, which is not possible for e.g. wireless transmissions.
+
+In a digital system, we could understand \acrshort{snr} as the probability of a bit being flipped in transit.
+For example, if a binary message is sent electronically, with a 1 being represented by a voltage of one volt
+and a 0 being represented by zero volts,
+the receiver is likely to observe some intermediate value such as .82 volts.
+Using a nearest-neighbor criterion, the receiver might consider anything over .5 volts to be the binary signal 1
+and anything less than .5 volts to be the binary signal 0.
+If the noise level is small on average, most transmissions will be interpreted correctly.
+However, there generally remains a chance that an intended 1 will be received as .47 volts and hence interpreted incorrectly as a 0.
+\cite{enwiki:signal-to-noise}
+
+In this digital system, we can improve communication reliability by using a coding scheme that is tolerant of errors.
+
+\section{Hamming Condition}
+theorems: Hamming condition, Varsham-Gilbert
+Shannon-Hartley
+Convolutional Code
+Reed-Solomon
 CRC
 \printbibliography
 \end{document}