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docs/general_idea.md

@@ -1,6 +1,8 @@
 # Homotopy Continuation
 
-### Introduction
+## Introduction
+
+### Overview
 
 The aim of a homotopy method consists in solving a system of N nonlinear
 equations in N variables \[1, p.1\]:
@@ -85,6 +87,26 @@ between successive points produced by the iterations can be used as a criterion
 for convergence. Of course, if the iterations fail to converge, one must go
 back to adjust the step size for the Euler’s predictor." [2, p.130]
 
+## Application to Channel Decoding
+
+We can describe the decoding problem using the code constraint polynomial [3]
+
+$$
+h(\bm{x}) = \underbrace{\sum_{i=1}^{n}\left(1-x_i^2\right)^2}_{\text{Bipolar constraint}} + \underbrace{\sum_{j=1}^{m}\left(1 - \left(\prod_{i\in A(j)}x_i\right)\right)^2}_{\text{Parity constraint}},
+$$
+
+where $A(j) = \left\{i \in [1:n]: H_{j,i} = 1\right\}$ represents the set of
+variable nodes involved in parity check j. This polynomial consists of a set of
+terms representing the bipolar constraint and a set of terms representing the
+parity constraint. In a similar vein, we can define the following set of
+polynomial equations to describe codewords:
+
+$$
+F = \left[\begin{array}{c}1 - x_1^2 \\ \vdots\\ 1 - x_n^2 \\ 1 - \prod_{i \in A(1)}x_i \\ \vdots\\ 1 - \prod_{i \in A(m)}x_i\end{array}\right] \overset{!}{=} \bm{0}.
+$$
+
+This is a problem we can solve using homotopy continuation.
+
 ______________________________________________________________________
 
 ## References
@@ -97,3 +119,7 @@ Philadelphia, PA 19104), 2003. doi: 10.1137/1.9780898719154.
 \[2\]: T. Chen and T.-Y. Li, “Homotopy continuation method for solving systems
 of nonlinear and polynomial equations,” Communications in Information and
 Systems, vol. 15, no. 2, pp. 119–307, 2015, doi: 10.4310/CIS.2015.v15.n2.a1.
+
+\[3\]: Wadayama, Tadashi, and Satoshi Takabe. "Proximal decoding for LDPC
+codes." IEICE Transactions on Fundamentals of Electronics, Communications and
+Computer Sciences 106.3 (2023): 359-367.