## 3 Codewords Analysis
### Structured Codewords  
In [18], [PERSON] _et al_. introduced a method to calculate the deterministic codewords for quasi-cyclic (QC) LDPC block codes. In this section, we give the proof to show that the method can also be applied to LDPC-CCs. These deterministic codewords refer to the structured codewords in this paper. **Lemma 1**. : _Let \(\mathcal{C}\) be the LDPC-CC defined by the polynomial syndrome former matrix \(\textbf{H}^{T}(D)\) in Equation (1) and let \(S\) be a subset of \(\{0,1,2,\ldots,n-1\}\) with \(p+1\) elements. Arbitrarily taking \((p+1)\) rows of \(\textbf{H}^{T}(D)\) with the row indices in \(S\) forms the submatrix denoted by \(\textbf{H}^{T}_{S}(D)\). Let \(\textbf{W}(D)=[\textbf{w}^{(0)}(D),\textbf{w}^{(1)}(D),\cdots,\textbf{w}^{(n- 1)}(D)]\) be a vector defined by_  
\[\textbf{w}^{(i)}(D)\triangleq\left\{ \begin{array}{ll}\text{perm}(\textbf{H }^{T}_{S\backslash i}(D))&\text{if }\,i\in S\\ 0&\text{otherwise}\end{array} \right. \tag{7}\]  
_where \(S\backslash i\) indicates excluding \(i\) from the set \(S\), and \(\text{perm}(\cdot)\) calculates the involved permanent of the polynomial matrix. Then \(\textbf{W}(D)\) is a codeword in \(\mathcal{C}\)._  
Proof. : See the Appendix. **Definition 1**. : _A codeword calculated using Lemma 1 is defined as the base structured codeword._  
Since the codeword calculated in Lemma 1 is determined by the permanent of the polynomial submatrices, modifying the powers of the monomials in \(\textbf{H}^{T}(D)\) only changes the form of the codewords, but not the amount of them, therefore, we call them structured. Additionally, since the sum of any (periodically shifted) codewords obtained from Lemma 1 defines another structured codeword, hence we consider them as the base ones. Given the polynomial syndrome former matrix of size \(n\times p\), there are \(\binom{n}{p+1}\) base structured codewords. The minimum weight of these base structured codewords gives an upper bound on the free distance of the LDPC-CC. To illustrate the concept of the base structured codewords, the \((21,3,5)\) Tanner LDPC-CC of rate \(R=b/n=(n-p)/n=2/5\) is chosen as an example. This code is defined by the polynomial syndrome former matrix  
\[\textbf{H}^{T}(D)= \begin{bmatrix}1&1&D^{18}\\ D&D^{5}&D^{12}\\ D^{3}&D^{15}&1\\ D^{7}&D^{4}&D^{7}\\ D^{15}&D^{13}&D^{21}\end{bmatrix} _{5\times 3} \tag{8}\]  
which is obtained by removing the common factors of \(D\) from each column of \(\textbf{H}^{T}(D)\) from the \((31,3,5)\) code in [19] for simplicity. According to Lemma 1, we choose the submatrix of size \((p+1)\times p=4\times 3\) as follows  
\[ \begin{bmatrix}1&1&D^{18}\\ D&D^{5}&D^{12}\\ D^{3}&D^{15}&1\\ D^{7}&D^{4}&D^{7}\end{bmatrix} _{4\times 3} \tag{9}\]There are four different square submatrices of size \(3\times 3\) in Equation (9), and the permanent of each square submatrix is given in Table 1. According to Equation (7), the 5-tuple sequence  
\[\textbf{W}(D)=[\textbf{w}^{0}(D),\textbf{w}^{1}(D),\textbf{w}^{2}(D),\textbf{w}^{ 3}(D),\textbf{0}(D)] \tag{10}\]  
or, equivalently,  
\[\textbf{w}(D)= D\cdot\textbf{w}^{0}(D^{5})+D^{2}\cdot\textbf{w}^{1}(D^{5})+D^{3 }\cdot\textbf{w}^{2}(D^{5})+D^{4}\cdot\textbf{w}^{3}(D^{5})+D^{5}\cdot\textbf{ 0}(D^{5})\] \[= (D^{4}+D^{17}+D^{21}+D^{24}+D^{32}+D^{38}+D^{47}+D^{56}+D^{58}+D^{ 71}+D^{74}\] \[+D^{78}+D^{91}+D^{93}+D^{107}+D^{111}+D^{113}+D^{122}+D^{129}\] \[+D^{134}+D^{148}+D^{166}+D^{169}+D^{197})\cdot D^{5}\]  
is a base structured codeword of Equation (8), and it satisfies the parity-check constraint \(\textbf{W}(D)\textbf{H}^{T}(D)=\textbf{0}(D)\). The polynomial syndrome former matrix \(\textbf{H}^{T}(D)\) in Equation (8) contains five submatrices of size \(4\times 3\), and each one defines a base structured codeword. All of these five base structured codewords are presented in Table 2, and each one is of weight 24. Therefore, the free distance of the (21,3,5) Tanner LDPC-CC is upper bounded by 24 which is consistent with the estimated free distance in [17].