# Soft Decision Decoding with Cyclic Information Set and the Decoder Architecture for Cyclic Codes
## 1 Introduction  
Interest in short block length codes has been rising recently, mainly due to emerging applications that require high reliability and low latency data transmission. Examples of such applications are machine-type communications in 5G/6G, the Internet of Things, smart metering networks, remote command links, and messaging services [1; 2; 3; 4; 5; 6; 7; 8]. Short block length codes play a crucial role in satisfying the low latency requirements of 5G communication. However, for short block length codes, the necessary trade-off between good error correction performance and decoding complexity leads to a significant gap between error correction performance and the theoretical lower bound, and on the other hand, leads to a dearth of practical decoding solutions [9; 10]. The cyclic codes, first proposed in the 1960s [11], can be soft decision decoded based on their code structure such as the trellis graph or the bit reliability [12; 13; 14; 15; 16]. Decoding algorithms based on the trellis graphs can approach the maximum likelihood (ML) soft decision decoding performance, but the decoding complexity is the main concern, especially when the code length \(n\) is not so small and the states grow exponentially with \(k\) or \(n-k\), where \(k\) is the information bit number [12; 13]. The generalized minimum distance decoding has a relatively low decoding complexity. However, the gap to the performance of the ML decoding could not be neglected [14; 15]. The decoding schemes based on ordered statistics can approach ML performance and have recently become a hot topic, showing great potential in future wireless communications for short or moderate length codes [16; 17]. The ordered statistic decoding (OSD) algorithm was proposed by [PERSON] and [PERSON] in [16]. The traditional algorithms based on ordered statistics need to find the most reliable basis(MRB), which involves sorting the soft information of all the bits and finding \(k\) independent columns of the generation matrix associated with the \(k\) most reliable bits [16]. In order to eliminate the burden of identifying the MRB, many efforts have been paid to reduce the complexity and some significant progress has been achieved [17]. The main ideas of the existing improved OSD algorithms include skipping and/or discarding unlikely test patterns to avoid unnecessary calculations [18; 19; 20; 21; 22; 23; 24; 25; 26]. A double re-encoding technique was proposed in [27]. Two groups of independent candidates are chosen for a particular class of rate-1/2 systematic block codes. Without using the MRB, this method only needs a few candidate codewords, which is practical for hardware implementation [28; 29]. The application of the OSD algorithms for short codes in wireless communications, especially the ultra-reliable and low latency communication (URLLC), has also been investigated [30; 31; 32]. Decoder architectures of all kinds of decoding algorithms, including the soft decision decoding algorithms, have also been comprehensively evaluated. As a classical example, the Golay code is popular for its perfect structure and good error correction capability. Many important decoding algorithms have been proposed for Golay codes in various application scenarios [33; 34; 35; 36; 37; 38]. Besides the traditional URLLC applications, the code was also applied in some other new scenarios [39; 40]. There are many decoder examples for Golay codes [41; 42; 43; 44]. For example, [PERSON] et al. proposed a decoder for extended binary Golay codes based on the incomplete ML decoding scheme with low latency and low complexity [41]. [PERSON] et al. exploited the properties of the Golay code to implement a parallel decoder that corrects single and double-adjacent errors [42]. Short BCH codes are another type of important block code suitable for soft decision decoding. Their implementation architectures have also been proposed [45; 46; 47; 48; 49].