## 1 Introduction  
Low-density parity-check block codes (LDPC-BCs) have been widely used in many communication standards, such as WiMax, DVB2, and 802.11n [1; 2; 3]. As the counterpart of LDPC-BCs, LDPCconvolutional codes (LDPC-CCs), also known as specially-coupled LDPC codes, were first proposed in [4] and have been extensively investigated in [5; 6; 7; 8; 9]. For the design of practical LDPC-CCs, it is preferred to have relative large girth (the length of the shortest cycle) and large free distance (the minimum weight of the codewords), in this paper, we focus on the analysis of the low weight codewords of LDPC-CCs.  
Distance spectrum can be described by a set \(\{A_{w}|w\geq d_{f},w\in\mathbb{Z}^{+}\}\), where \(A_{w}\), the codeword weight enumerator, is the number of codewords with Hamming weight \(w\), and the free distance \(d_{f}\) is defined as the minimum weight of the codewords. Under linear programming decoding, the low weight pseudocodewords of LPDC codes were analysed in [10; 11]. The pseudocodeword performance of an LDPC-CC and the underlying quasi-cyclic (QC) code was presented in [12]. Regarding the calculation of the distance spectrum of conventional convolutional codes, some work has been done in [13; 14; 15], and [PERSON] _et al_. [16] presented a more efficient bidirectional tree search algorithm called BEAST. Compared to the conventional convolutional codes, LDPC-CCs have relative larger memory order because of the sparsity of the syndrome former matrix. Instead of calculating the precise distance spectrum, [PERSON] _et al_. [17] introduced a method to estimate the distance spectrum of LDPC-CCs by evaluating the linear dependence between the low weight codewords of super codes. A super code is defined by the submatrix obtained by splitting the polynomial-domain syndrome former matrix of the mother code.  
Rather than calculating or estimating the distance spectrum of LDPC-CCs. In this paper, we introduce a method to improve the distance spectrum of LDPC-CCs by eliminating the so-called non-structured low weight codewords. The codewords of polynomial-based LDPC-CCs are characterised into two categories, _i.e._, structured and non-structured. In the distance spectrum, the number of the structured codewords is dominated by the size of the polynomial syndrome former matrix \(\textbf{H}^{T}(D)\), while the number of the non-structured ones depends on the particular monomials or polynomials in \(\textbf{H}^{T}(D)\).  
By evaluating the relationship of the codewords between the mother code and its super codes, we find that the structured codeword of the mother code can only be derived from the structured codewords of a super code, while the non-structured one of the mother code requires both structured and non-structured super codewords to convert. Therefore, as the main contribution of the paper, eliminating low weight non-structured codewords in the super codes by appropriately choosing the monomials in \(\textbf{H}^{T}(D)\) can decrease the number of low weight non-structured codewords in the mother code, resulting in improved distance spectrum.  
The rest of the paper is organised as follows: Section 2 gives the definition of the polynomial-domain LDPC-CCs. The concept of structured and non-structured codewords is discussed in Section 3. Section 4 demonstrates the approach of eliminating low weight non-structured codewords, and gives the designing criteria of practical LDPC-CCs with respect to good distance spectrum. Section 5 concludes the work.