# Minimum Distance Optimization with Chord Edge Growth for High Girth Non-Binary LDPC Codes
## 1 Introduction  
The design of efficient short codes has attracted widespread interest due to the emergence of applications requiring high reliability and low latency data transmission [1, 2]. The low-density parity check (LDPC) codes, which were first proposed by [PERSON] et al. [3] and rediscovered by [PERSON] and [PERSON] [4, 5], have performance close to the Shannon limit as the code length approaches infinity [6]. For short or moderate code lengths, non-binary low-density parity-check (NB-LDPC) codes exhibit better error-correction capabilities than binary LDPC codes in certain cases [7, 8]. The construction of NB-LDPC codes can adopt algebraic or graph theory methods. The algebraic methods for constructing non-binary quasi-cyclic LDPC codes that are based on finite Euclidean geometric planes and array masks were studied by [PERSON] et al. [9], [PERSON] et al. [10] and [PERSON] et al. [11]. The codes that were constructed by these methods can be effectively encoded and they have excellent performance on the Additive White Gaussian Noise (AWGN) channel. The design methods of non-binary protograph-based LDPC codes with excellent performance were studied by [PERSON] et al. [12], [PERSON] et al. [13] and [PERSON] et al. [14]. The design of an LDPC code with excellent performance in the high signal-to-noise ratio (SNR) region is challenging due to the presence of the error floor. It has been found that the error floor is largely attributed to the effect of error-prone graph structures in the Tanner graph of the code, such as stopping sets and trapping sets [15, 16, 17]. Therefore, much work has been devoted to theconstruction of NB-LDPC codes that can avoid these harmful substructures [14; 15; 16; 18; 19]. In order to avoid small absorbing sets, a design guide of NB-LDPC codes with the column weight greater than two was proposed by [PERSON] et al. [16], which can significantly reduce the error floor of the graph-based non-binary codes. Because these small substructures are related to short cycles, the error floor can be mitigated by manipulating the parameters of the cycles [14; 15]. The designing scheme of the NB-LDPC codes by maximizing the quasi girth and the minimum distance of the subgraph was studied by [PERSON] et al. [14]. However, this method needs to continuously expand the tree to determine the local girth and calculate the minimum distance upper limit, which increases the complexity of the algorithm. The ultra-sparse NB-LDPC codes with the column weight of two defined on the high-order finite field or general linear group have excellent performance at finite length [7; 8; 15; 20]. For the optimization of this type of ultra-sparse NB-LDPC codes, a method that is based on the algebraic properties of the binary image of the parity-check matrix that is associated with the code to ensure the performance of the waterfall region and the error floor region was introduced by [PERSON] et al. [15], where the algebraic properties of the cycles and the stopping sets were linked to the global minimum distance of the code. However, it does not take advantage of the characteristics of ultra-sparse codes to simplify the optimization method for the numerous cycles in the corresponding graph. Because ultra-sparse codes can be described by simple graphs [21], the sub-optimal optimization methods combining the characteristics of graphs can be used to design the NB-LDPC codes with the outstanding performance efficiently. Based on the studies by [PERSON] et al. [22; 23; 24], the design method can be further modified to obtain the NB-LDPC code that can be encoded by a simple recursive method, thereby achieving efficient encoding. In this paper, a progressive chord edge growth strategy (PCEG) for ultra-sparse NB-LDPC codes with high girth, which can effectively lower the error floor of NB-LDPC codes, is proposed. The chord edges are added to the Hamiltonian cycle in turn, and the candidate non-zero entries corresponding to each chord edge are searched.