## 3 Codewords Analysis
### Non-Structured Codewords  
Unlike the structured codewords, there is another category of codewords depending on the particular monomials or polynomials in \(\mathbf{H}^{T}(D)\). By appropriately choosing the power of each monomial, this type of codewords can be eliminated. We call them non-structured codewords in this paper. Note that, if any non-structured codewords are involved in the sum of a set of codewords, then the sum results in another non-structured codeword. In [17], a method was introduced to estimate the distance spectrum of LDPC-CCs (mother code) by splitting the polynomial syndrome former matrix into submatrices representing \"super codes\" and then evaluating the linear dependence between codewords of the corresponding super codes. We apply this method to the (21,3,5) Tanner LDPC-CC. First, the polynomial syndrome former matrix of the original code is split into two submatrices \(\mathbf{H}^{T}_{a}(D)\) and \(\mathbf{H}^{T}_{b}(D)\) as follows  
\[\mathbf{H}^{T}_{a}(D)= \begin{bmatrix}1&1\\ D&D^{5}\\ D^{3}&D^{15}\\ D^{7}&D^{4}\\ D^{15}&D^{13}\end{bmatrix} ,\mathbf{H}^{T}_{b}(D)= \begin{bmatrix}1&D^{18}\\ D^{5}&D^{12}\\ D^{15}&1\\ D^{4}&D^{7}\\ D^{13}&D^{21}\end{bmatrix} . \tag{11}\]  
Each one defines a super code and has free distance of 6. By computer search, there are 22 and 12 codewords of weight 6 in the set \(\{\mathbf{v}^{i}_{a}(D)|i=1,2,\ldots,22\}\) and \(\{\mathbf{v}^{j}_{b}(D)|j=1,2,\ldots,12\}\) for \(\mathbf{H}^{T}_{a}(D)\) and \(\mathbf{H}^{T}_{b}(D)\), respectively. In addition, no codewords of weight smaller than six are found, therefore, the free distance of these two super codes are six. Applying Lemma 1 to \(\mathbf{H}^{T}_{a}(D)\) and \(\mathbf{H}^{T}_{b}(D)\), as shown in \begin{table}
\begin{tabular}{l l} \hline  & \(D^{4}+D^{17}+D^{21}+D^{24}+D^{32}+D^{38}+D^{47}+D^{56}+D^{58}+\) \\ \(\mathbf{v}_{1}(D)\) & \(D^{71}+D^{74}+D^{78}+D^{91}+D^{93}+D^{107}+D^{111}+D^{113}+\) \\  & \(D^{122}+D^{129}+D^{134}+D^{148}+D^{166}+D^{169}+D^{197}\) \\ \hline  & \(D^{5}+D^{20}+D^{35}+D^{49}+D^{59}+D^{76}+D^{81}+D^{83}+D^{93}+\) \\ \(\mathbf{v}_{2}(D)\) & \(D^{95}+D^{96}+D^{104}+D^{108}+D^{110}+D^{121}+D^{123}+D^{154}+\) \\  & \(D^{164}+D^{166}+D^{168}+D^{173}+D^{185}+D^{196}+D^{224}\) \\ \hline  & \(D^{5}+D^{25}+D^{45}+D^{60}+D^{62}+D^{66}+D^{72}+D^{74}+D^{80}+\) \\ \(\mathbf{v}_{3}(D)\) & \(D^{87}+D^{89}+D^{91}+D^{94}+D^{96}+D^{99}+D^{102}+D^{115}+\) \\  & \(D^{116}+D^{121}+D^{124}+D^{126}+D^{147}+D^{152}+D^{154}\) \\ \hline  & \(D^{5}+D^{40}+D^{49}+D^{55}+D^{72}+D^{75}+D^{77}+D^{79}+D^{83}+\) \\ \(\mathbf{v}_{4}(D)\) & \(D^{92}+D^{95}+D^{108}+D^{113}+D^{117}+D^{119}+D^{124}+D^{133}+\) \\  & \(D^{138}+D^{143}+D^{150}+D^{162}+D^{164}+D^{189}+D^{192}\) \\ \hline  & \(D^{5}+D^{25}+D^{62}+D^{66}+D^{72}+D^{75}+D^{96}+D^{108}+D^{117}+\) \\ \(\mathbf{v}_{5}(D)\) & \(D^{123}+D^{128}+D^{130}+D^{133}+D^{135}+D^{136}+D^{141}+D^{158}+\) \\  & \(D^{167}+D^{170}+D^{177}+D^{181}+D^{188}+D^{206}+D^{237}\) \\ \hline \end{tabular}
\end{table} 
Table 2: The base structured codewords of the \((21,3,5)\) Tanner LDPC-CC. Table 3 the codewords \(\{\textbf{v}_{a}^{k}(D)|k=1,2,\ldots,10\}\) and \(\{\textbf{v}_{b}^{l}(D)|l=1,2,\ldots,10\}\) are the base structured codewords, and the rest are the non-structured ones. By evaluating the linear dependence between the minimum weight codewords of the super codes in Table 3, we find that the codewords of the mother code in Table 2 can be derived from each super code by summing some periodically shifted base structured codewords. For example,  
\[\left\{ \begin{array}{l}\textbf{v}_{1}(D)=D^{5\cdot 22}\textbf{v}_{a}^{1}(D)+D^{5 \cdot 7}\textbf{v}_{a}^{2}(D)+D^{5\cdot 0}\textbf{v}_{a}^{3}(D)+D^{5\cdot 14} \textbf{v}_{a}^{4}(D)\\ \textbf{v}_{1}(D)=D^{5\cdot 9}\textbf{v}_{b}^{1}(D)+D^{5\cdot 3}\textbf{v}_{b}^{2}(D)+D ^{5\cdot 6}\textbf{v}_{b}^{3}(D)+D^{5\cdot 0}\textbf{v}_{b}^{4}(D)\end{array}