Patent Publication Number: US-10320522-B2

Title: Packet encoding and decoding method and apparatus

Description:
PRIORITY 
     This application is a continuation application of prior application Ser. No. 14/105,930, filed on Dec. 13, 2013, which claimed the benefit under 35 U.S.C. § 119(a) of a Korean patent application filed on Dec. 14, 2012 in the Korean Intellectual Property Office and assigned Serial number 10-2012-0146579, the entire disclosure of which is hereby incorporated by reference. 
    
    
     JOINT RESEARCH AGREEMENT 
     The present disclosure was made by or on behalf of the below listed parties to a joint research agreement. The joint research agreement was in effect on or before the date the present disclosure was made and the present disclosure was made as a result of activities undertaken within the scope of the joint research agreement. The parties to the joint research agreement are 1) SAMSUNG ELECTRONICS CO., LTD. and 2) SUNGKYUNKWAN UNIVERSITY RESEARCH &amp; BUSINESS FOUNDATION. 
     TECHNICAL FIELD 
     The present disclosure relates to a data packet decoding method and apparatus. More particularly, the present disclosure relates to a method and an apparatus for encoding and decoding packets using a polar code. 
     BACKGROUND 
     It is imperative to transmit information without any loss in data communication. In a wireless communication system, however, a radio signal carrying information is distorted due to noises, multipath fading, interferences, and the like. Therefore, there have been many studies on error-correcting codes to improve signal reception reliability with the addition of well-controlled redundant information. 
     The polar code has been proposed first in 2008. The polar code is characterized by low coding and decoding complexity. The polar code is the first error correction code proved to be able to achieve the Shannon&#39;s channel capacity as a theoretical limit on the general Binary-input Discrete Memoryless symmetric Channel (B-DMC). 
     Meanwhile, the Successive Cancellation (SC) decoder proposed to decode the polar code has shown inferiority as compared to the Low-Density Parity Check (LDPC) code and Turbo code in SC decoding performance on the polar code having a finite code length N. Recently, a Successive Cancellation List (SCL) decoder has been proposed in order to overcome this performance inferiority. 
     The SCL decoder is an expanded SC decoder so as to decode the message bits successively through successive cancellation like the SC decoder. However, unlike the SC decoder having one decoding path, the SCL decoder has L decoding paths that are managed in a list and selects a codeword corresponding to one of L decoding paths. The codeword selection is performed under the rule of selecting the codeword having the highest posterior probability. 
       FIG. 1  is a graph illustrating a polar code decoding performance of an SCL decoder according to the related art. 
     Referring to  FIG. 1 , the horizontal axis denotes channel quality ((E b /N 0 ) and the vertical axis denotes the Bit-Error Rate (BER).  FIG. 1  shows an error floor region  110  and a waterfall region  120 . 
     Referring to  FIG. 1 , the decoding performance of the SCL decoder increases as the size L of the SCL decoder increases. As shown in  FIG. 1 , however, the decoding performance of the SCL decoder on the polar code having the finite code length N shows the error floor region  110 . This is because the linear code generated according to the normal polar code generation method has a relatively short minimum distance. 
       FIG. 2  is a block diagram illustrating a configuration of a Cyclic Redundancy Check (CRC)-polar code concatenation encoder according to the related art. 
     Referring to  FIG. 2 , as one of the approaches to address the aforementioned issue of the SCL decoder, a method of concatenating a CRC code and a polar code has been proposed. A CRC-α coder  220  is a kind of error detection code. A message  210  is CRC-coded by the CRC-α coder  220  and polar-coded by a polar coder  230 . Such coding operations are performed by a CRC-polar concatenation encoder  240 . 
       FIG. 3  is a graph illustrating decoding performance of an encoder according to the related art. 
     Referring to  FIG. 3 , the horizontal axis denotes the channel quality (E b /N 0 ) and the vertical axis denotes the BER. 
     As shown in  FIG. 3 , the error floor region is overcome with the concatenation of the polar code and the CRC code. 
     The CRC code assists the SCL decoder to select a codeword corresponding to one of L decoding paths as the decoding results of the SCL decoder. The SCL decoder implemented by concatenating the CRC code with the polar coder removes the codewords that failed to pass the CRC test and selects the codeword having the highest probability among the codewords that passed the CRC test. 
     Therefore, a need exists for a method and an apparatus for efficiently encoding and decoding packets using a polar code. 
     The above information is presented as background information only to assist with an understanding of the present disclosure. No determination has been made, and no assertion is made, as to whether any of the above might be applicable as prior art with regard to the present disclosure. 
     SUMMARY 
     Aspects of the present disclosure are to address at least the above-mentioned problems and/or disadvantages and to provide at least the advantages described below. Accordingly, an aspect of the present disclosure is to provide an efficient encoding and decoding method and apparatus. 
     In accordance with an aspect of the present disclosure, a method for decoding a packet including multiple blocks is provided. The method includes acquiring a plurality of blocks constituting the packet, extracting a plurality of codeword candidates corresponding to the blocks, selecting some of the plurality of codeword candidates in a descending order of posterior probability among the plurality of codeword candidates corresponding to the blocks, combining the selected codeword candidates into a plurality of codeword combinations, selecting a codeword combination having a highest posterior probability and passed Cyclic Redundancy Check (CRC) test without error among the plurality of codeword combinations, and decoding the selected codeword combination. 
     In accordance with another aspect of the present disclosure, an apparatus for decoding a packet including multiple blocks is provided. The apparatus includes a communication unit configured to receive a plurality of blocks constituting the packet and a control unit configured to acquire the plurality of blocks constituting the packet, to extract a plurality of codeword candidates corresponding to the blocks, to select some of the plurality of codeword candidates in a descending order of posterior probability among the plurality of codeword candidates corresponding to the blocks, to combine the selected codeword candidates into a plurality of codeword combinations, to select a codeword combination having a highest posterior probability and passed CRC test without error among the plural codeword combinations, and to decode the selected codeword combination. 
     Other aspects, advantages, and salient features of the disclosure will become apparent to those skilled in the art from the following detailed description, which, taken in conjunction with the annexed drawings, discloses various embodiments of the present disclosure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other aspects, features, and advantages of certain embodiments of the present disclosure will be more apparent from the following description taken in conjunction with the accompanying drawings, in which: 
         FIG. 1  is a graph illustrating a polar code decoding performance of a Successive Cancellation List (SCL) decoder according to the related art; 
         FIG. 2  is a block diagram illustrating a configuration of a Cyclic Redundancy Check (CRC)-polar code concatenation encoder according to the related art; 
         FIG. 3  is a graph illustrating decoding performance of an encoder according to the related art; 
         FIG. 4A  is a diagram illustrating a segmentation of data into packets according to an embodiment of the present disclosure; 
         FIG. 4B  is a diagram illustrating a principle of an encoding method according to an embodiment of the present disclosure; 
         FIG. 4C  is a diagram illustrating a decoding procedure according to an embodiment of the present disclosure; 
         FIG. 5A  is a diagram illustrating a principle of an encoding method according to an embodiment of the present disclosure; 
         FIG. 5B  is a flowchart illustrating an encoding method according to an embodiment of the present disclosure; 
         FIG. 6  is a flowchart illustrating a decoding procedure according to a first embodiment of the present disclosure; 
         FIG. 7  is a diagram illustrating a decoding principle according to the first embodiment of the present disclosure; 
         FIG. 8  is a diagram illustrating a principle of a decoding procedure according to the first embodiment of the present disclosure; 
         FIG. 9  is a flowchart illustrating a polar code decoding procedure according to a second embodiment of the present disclosure; 
         FIG. 10  is a diagram illustrating a procedure of decoding a polar code according to the second embodiment of the present disclosure; 
         FIG. 11  is a diagram illustrating a principle of a decoding procedure according to the second embodiment of the present disclosure; 
         FIGS. 12, 13, and 14  are graphs illustrating performance comparison results between the method of the related art and the proposed method according to an embodiment of the present disclosure; 
         FIG. 15  is a flowchart illustrating a procedure for decoding a packet comprising a plurality of blocks according to a second embodiment of the present disclosure; and 
         FIG. 16  is a diagram illustrating an apparatus for decoding a packet comprising a plurality of blocks, according to an embodiment of the present disclosure. 
     
    
    
     Throughout the drawings, it should be noted that like reference numbers are used to depict the same or similar elements, features, and structures. 
     DETAILED DESCRIPTION 
     The following description with reference to the accompanying drawings is provided to assist in a comprehensive understanding of various embodiments of the present disclosure as defined by the claims and their equivalents. It includes various specific details to assist in that understanding but these are to be regarded as merely exemplary. Accordingly, those of ordinary skill in the art will recognize that various changes and modifications of the various embodiments described herein can be made without departing from the scope and spirit of the present disclosure. In addition, descriptions of well-known functions and constructions may be omitted for clarity and conciseness. 
     The terms and words used in the following description and claims are not limited to the bibliographical meanings, but, are merely used by the inventor to enable a clear and consistent understanding of the present disclosure. Accordingly, it should be apparent to those skilled in the art that the following description of various embodiments of the present disclosure is provided for illustration purpose only and not for the purpose of limiting the present disclosure as defined by the appended claims and their equivalents. 
     It is to be understood that the singular forms “a,” “an,” and “the” include plural referents unless the context clearly dictates otherwise. Thus, for example, reference to “a component surface” includes reference to one or more of such surfaces. 
     By the term “substantially” it is meant that the recited characteristic, parameter, or value need not be achieved exactly, but that deviations or variations, including for example, tolerances, measurement error, measurement accuracy limitations and other factors known to skill in the art, may occur in amounts that do not preclude the effect the characteristic was intended to provide. 
     Some elements are exaggerated, omitted, or simplified in the drawings and the elements may have sizes and/or shapes different from those shown in drawings, in practice. The same reference numbers are used throughout the drawings to refer to the same or like parts. 
     An embodiment of the present disclosure is directed to a method for decoding polar codewords concatenated in a unit of packets using a Successive Cancellation List (SCL) decoder. The encoder/decoder uses a packet Cyclic Redundancy Check (CRC) code concatenated to the packet in CRC decoding instead of the CRC code concatenated to the polar codeword in a unit of blocks. The decoding method is capable of reducing the coding rate with the CRC code concatenated to each block, resulting in improvement of polar code decoding performance. 
     The packet decoding methods according to various embodiments of the present disclosure can be applied to various communication systems transmitting data in a unit of packets. Each packet is split into blocks and polar-coded to be transmitted to the recipient. The encoder/decoder according to an embodiment of the present disclosure removes the CRC code concatenated to each block to prevent coding rate loss so as to improve the packet decoding performance. The encoder does not concatenate the CRC code to each block in the coding process. The decoder does not perform codeword decoding on every block in the decoding procedure using the SCL decoder but leaves codeword candidates in size M so as to improve packet decoding performance using combinations of the codeword candidates. 
     A decoder which decodes codewords concatenated in a unit of packets. The large size information is split into multiple data units for facilitating transmission and such a data unit is called a packet. At this time, a CRC code is concatenated to each packet in order for the recipient to determine whether the received data is erroneous. In wireless data communication, when encoding a large packet, the packet is fragmented into multiple blocks which are coded respectively and then concatenated back together. 
       FIG. 4A  is a diagram illustrating a segmentation of data into packets according to an embodiment of the present disclosure. 
       FIG. 4B  is a diagram illustrating a principle of an encoding method according to an embodiment of the present disclosure. 
     Referring to  FIGS. 4A and 4B , information data  410  is segmented into packets  420  and  440 , and each packet  420  is split into four blocks  422 ,  425 ,  428 , and  431  which are coded by respective polar coders  424 ,  427 ,  430 , and  433  and transmitted. The entire method is divided into a polar code encoding and decoding procedures. 
     Polar Code Encoding Procedure 
     The information data  410  to be transmitted by the transmitter is fragmented into multiple packets  420  and  440 . A CRC code  421  is added to each fragment to form a packet  420 . Each packet is segmented into a plurality of data blocks. For example, the packet  420  is split into the blocks  422 ,  425 ,  428 , and  431 . The CRC codes  423 ,  426 ,  429 , and  432  are concatenated to the segments to form the blocks  422 ,  425 ,  428 , and  431 . The CRC codes  423 ,  426 ,  429 , and  432  are used for maximizing the decoding performance of the polar code with the SCL decoder. The blocks  422 ,  425 ,  428 , and  431  having the respective CRC codes  423 ,  426 ,  429 , and  432  are polar coded by the respective polar coder  424 ,  427 ,  430 , and  433  and transmitted to the receiver. The CRC codes  421  and  441  added to the respective packets are referred to as packet CRC codes, and the CRC codes  423 ,  426 ,  429 , and  432  added to the respective blocks are referred to as block CRC codes. 
     Polar Code Decoding Procedure 
       FIG. 4C  is a diagram illustrating a decoding procedure according to an embodiment of the present disclosure. 
     The packets  420  and  440  constituting the information data  410  are received by the receiver distorted due to the noise, multipath fading, interference, and the like. Accordingly, the signal carrying the packets is recovered through a decoding process. The received signal is decoded by using the SCL decoder. The SCL decoder generates L codeword candidates  452  to the signal vector (that is, a polar code block  451 ) received in a unit block. Among L codeword candidates  452 , the codeword candidates in which any error is detected through CRC test with the block CRC code are removed. Among the codeword candidates that passed the CRC test, one codeword having the highest posterior probability (that is, probability in which the corresponding codeword is likely to be the correct codeword) is selected. The methods of obtaining the posterior probability have been well-known. Each packet is recovered by concatenating the codewords decoded from blocks  451 ,  454 , and  455 . The receiver detects for an error on the concatenated codewords using the packet CRC code. If an error is detected at a certain phase of the decoding procedure, the receiver sends the transmitter a Negative acknowledgement (NACK) to request for retransmission and, otherwise, an Acknowledgement (ACK). 
     Referring to  FIG. 4C , a packet consists of K blocks. The blocks  451 ,  454 , and  455  are the polar code blocks received by the receiver. The polar code blocks  451 ,  454 , and  455  have the block CRC codes respectively. Each of the polar code blocks  451 ,  454 , and  455  represents L codeword candidates  452  decoded by the SCL decoder, the codeword candidates  452  being depicted right below the corresponding polar code block. For example, the codeword candidates  1   k  to L k  below the k th  block are the codeword candidates corresponding to the k th  block. The L codeword candidates  452  are arranged from top to bottom in the highest posterior-probability first order. For example, the codeword  2   1  denotes the codeword having the second highest posterior probability among the codeword candidates  452  of the polar code block  1   451 . The shaded part, such as a codeword  1   1    453  of the polar code block  1   451 , denotes the codeword selected as the transmitted codeword, and the non-shadowed part, such as codewords  2   1  to L 1 , denotes the codewords that are not selected. Each block is decoded in such a way of selecting the codeword having the highest posterior probability among the codeword candidates that passed the CRC test without error. The codewords  453 ,  456 , and  457  decoded from the respective blocks in this way are concatenated for a packet as shown in  FIG. 5A . The receiver detects for an error of the packet using the packet CRC code. 
     The technique of concatenating the CRC code in performing polar coding on the blocks of the data packet shows the best decoding performance in SCL decoding from the view point of a signal block. However, concatenating the CRC code to the individual blocks causes coding rate loss and thus, causes performance degradation from the view point of entire decoding performance. Since the packet CRC code is used for determining packet data integrity, this is also one of the causes of performance degradation. Therefore, there is a need of a more efficient encoding and decoding method. 
     The packet-based concatenation polar code decoding method according to an embodiment of the present disclosure is capable of performing SCL decoding with the packet CRC code instead of block CRC code concatenated to individual blocks which causes coding rate loss. 
     Polar Code Encoding Procedure 
       FIG. 5A  is a diagram illustrating a principle of an encoding method according to an embodiment of the present disclosure. 
       FIG. 5B  is a flowchart illustrating an encoding method according to an embodiment of the present disclosure. 
     Referring to  FIGS. 5A and 5B , the data to be transmitted by the transmitter is fragmented input multiple packets, such as packet  510  illustrated in  FIG. 5A , and each packet includes a corresponding CRC code, such as CRC code  511  illustrated in  FIG. 5A . Each packet is segmented into a plurality of blocks. Each block is polar-coded by the polar coder without concatenation of any block CRC code. For example, each block has no corresponding CRC code. The coded blocks are transmitted to the receiver. 
     Referring to  FIG. 5B , the transmitter (encoder) concatenates the packet CRC code  511  to the data to generate the packet  510  at operation  570 . The transmitter splits the packet  510  into K blocks  520 ,  530 ,  540 , and  550  at operation  572 . K denotes a number of blocks constituting the packet. In the case of  FIG. 5A , K=4. 
     Through operations  574 ,  576 ,  578 , and  580 , the transmitter performs polar coding on the individual blocks  520 ,  530 ,  540 , and  550 . Although the description is directed to the case of polar coding, the present disclosure is applicable to coding schemes without departing from the scope of the present disclosure. At operation  574 , i is set to 1 (i=1). The transmitter performs polar coding on the i th  block at operation  576 . The transmitter determines whether i is equal to K (i=K) at operation  578 . For example, the transmitter determines whether all of the blocks have been polar-coded. If not all of the blocks have been polar-coded, the transmitter increments i by 1 at operation  580 . Afterward, the transmitter repeat operations  576 ,  578 , and  580  until all of the blocks are polar-coded, i.e., until i=K. 
     Referring to  FIG. 5A , the first block  520  is polar-coded by the polar coder  525 , the second block  530  by the polar coder  535 , the third block  540  by the polar coder  545 , and the fourth block  550  by the polar coder  555 . 
     Although  FIG. 5B  is directed to the case where the blocks are polar coded in series, some or all of the blocks may be coded simultaneously or independently. 
     The encoding method depicted in  FIGS. 5A and 5B  corresponds to the decoding method to be described with reference to  FIGS. 6, 7, 8, 9, 10, and 11 . 
     The packet encoder performing the encoding procedure as described with  FIGS. 5A and 5B  may include a communication unit and a control unit. The communication unit may transmit the blocks constituting the packet according to the encoding procedure described with reference to  FIGS. 5A and 5B  under the control of the control unit. The control unit controls the encoding apparatus to encode/generate the blocks to be transmitted according to the encoding procedure of  FIGS. 5A and 5B . 
     In the present disclosure, two embodiments of the decoding procedure are proposed, and the proposed decoding procedures may operate independently. 
     Polar Code Decoding Procedure According to a First Embodiment 
       FIG. 6  is a flowchart illustrating a decoding procedure according to the first embodiment of the present disclosure. 
       FIG. 7  is a diagram illustrating a decoding principle according to the first embodiment of the present disclosure. 
     The packets carrying data is distorted by noise, multipath fading, interference, and the like, on the propagation channel to the receiver. In order to decode the signal correctly, the signal needs to be recovered. The received signal recovery procedure is described hereinafter. 
     Referring to  FIGS. 6 and 7 , the receiver (decoder) decodes the received signal by using the SCL decoder at operation  605 . The SCL decoder processes the received signal vectors, i.e., polar code blocks  710  and  720 , to generate L codeword candidates per polar code block. The receiver selects/acquires M codeword candidates having the highest posterior probability among the L codeword candidates per block. Here, M is an integer less than L. In the embodiment of  FIG. 7 , L=4 and M=2. In the embodiment of  FIG. 7 , the L codeword candidates acquired from the polar code block  1   710  include codeword  1   1 , codeword  2   1 , codeword  3   1 , and codeword  4   1 . The receiver selects M codewords (i.e., codeword  1   1  and codeword  2   1 ) among these codeword candidates. Similarly, in the embodiment of  FIG. 7 , the L codeword candidates acquired from the polar code block  2   720  include codeword  1   2 , codeword  2   2 , codeword  3   2 , and codeword  4   2 . The receiver selects M codewords (i.e., codeword  1   2  and codeword  2   2 ) among these codeword candidates. 
     The receiver sets k to 1 at operation  610 . The receiver generates combinations of codeword candidates of k th  and (k+i) th  blocks at operation  615 . The receiver determines whether k+1 is equal to the number of blocks constituting a packet at operation  620 . If it is determined at operation  620  that k+1 is less than the number of blocks constituting a packet, the receiver increment k by 1 at operation  625  and generates new combinations using the previously generated codeword combinations and the codeword candidates (M) of the (k−1) th  block at operation  630 . This process is repeated until k+1 reaches the number of blocks constituting a packet. In the embodiment of  FIG. 7 , K (that is, the number of blocks constituting a packet) is 2. Accordingly, single stage of k=1 is performed. 
     The receiver combines M codewords (i.e., codeword  1   1  and codeword  2   1 ) acquired from the polar code block  1   710  and M codewords (i.e., codeword  1   2  and codeword  2   2 ) acquired from the polar code block  2   720 . This generates four codeword combinations as denoted by reference number  730 . If M is identical in all of the blocks, the number of codeword combinations is M K . 
     If it is determined at operation  620  that k+1 reaches the number of blocks constituting a packet, the receiver performs error detection on the codeword combinations  730  using the packet CRC code at operation  635 . The receiver determines whether there is any codeword combination having no error at operation  640 . If it is determined at operation  640  that there is no codeword combination without error, the receiver selects all of the codeword combinations and the procedure goes to operation  650 . However, if it is determined at operation  640  that there is any codeword combination without error, the procedure goes to operation  645 . At operation  645 , the receiver selects the codeword combination(s) without error and discards the rest. At operation  650 , the receiver calculates the posterior probabilities of the codeword combinations and selects the codeword combination having the highest posterior probability. 
     According to an alternative embodiment, if there is no codeword combination without error at operation  640 , the receiver regards this as reception failure and sends the transmitter a NACK. Operations  635 ,  640 ,  645 , and  650  are the process of selecting a codeword combination having the highest posterior probability among the codeword combinations that passed the CRC test. Accordingly, it is possible to acquire the same result by performing CRC test on the codeword combinations in the descending order of posterior probability. 
     The posterior probability of the codeword combination is calculated as the product of the posterior probabilities of the combined codewords. For example, if the posterior probabilities of the combined codewords  1   1  and  2   2  are a and b respectively, the posterior probability of the codeword probability is a×b. The receiver performs decoding on the selected combination at operation  650  and, if the combination is decoded successfully, sends an ACK to the transmitter. 
     In the above embodiment, M is set arbitrarily at operation  605 . However, M may be set as follows depending on the embodiment. 
     P (codeword X k ) denotes the posterior probability of the codeword having X th  highest posterior probability in the codeword candidate list corresponding to k th  codeword block. The threshold value is expressed as Th. The size of M may be set differently or identically for the blocks. In the following, the description is directed to the case where the size of M is set differently for the blocks. M is equal to or less than L. However, M has to be set to a value less than L for at least one of the blocks. The receiver may set M to M1 if f(P( 1   t ), . . . , P(L t )) is equal to or greater than Th for the t th  block. In contrast, if f(P( 1   t ), . . . , P(L t ))&lt;Th, the receiver may set M to M2. Here, the function f( ) is the function having the code block of the corresponding block as input. 
     For example, f(P( 1   k ), . . . , P(L k ))=P( 1   k )−P( 2   k ). For example, if the difference between the highest and the second highest posterior probabilities among the posterior probabilities of the L codewords acquired from the k th  block is equal to or greater than the threshold value, M=M1. This means that M for the k th  block is set to M1 if P( 1   k )−P( 2   k )≥Th. In contrast, M for the k th  block may be set to M2=M−1+1 if P( 1   k )−P( 2   k )&lt;Th. The receiver may determine the size of the threshold value depending on the channel condition. 
     The embodiment of  FIG. 6  is directed to the case where the combination is performed in sequence. However, the combination may be performed in reverse order (i.e., in the order of K th , (K−1) th , . . . , first block). In the system capable of using multiple processors, it is possible to combine multiple pairs of blocks simultaneously, e.g., pair of the first and second blocks and pair of the third and fourth block. This alternative method may be applicable to the second embodiment described below. 
       FIG. 8  is a diagram illustrating a principle of a decoding procedure according to the first embodiment of the present disclosure. Although L, K, and M are set to small values to simplify the explanation in the embodiment of  FIG. 7 ,  FIG. 8  shows more general case as compared to the embodiment of  FIG. 7 . 
     Referring to  FIG. 8 , a packet consists of K blocks. The polar code block  1   810  is the first polar-coded block of the packet. The codewords  810 ,  820 , and  830  depicted below the respective blocks are L codeword candidates decoded by the SCL decoder. The L codeword candidates are arranged in the descending order of the posterior probability from top to bottom. For example, the codeword  2   1  is the codeword having the second highest posterior probability among the codeword candidates of the polar code block  1   810 . 
     For the polar code block  1   810 , M codewords ( 1   1 ,  2   1 , . . . , M 1 ) are selected. For the polar code block  2   830 , M codewords ( 1   2 ,  2   2 , . . . , M 2 ) are selected. At operation  615  of  FIG. 6 , the combinations of the codewords ( 1   1 ,  2   1 , . . . , M 1 ) and ( 1   2 ,  2   2 , . . . , M 2 ) are generated and denoted by reference number  840 . Thereafter, the combinations  840  and M codewords ( 1   3 ,  2   3 , . . . , M 3 ) corresponding to the polar code block  1   830  are combined and this process is repeated for K blocks capable of forming one packet. 
     Polar Code Decoding Procedure According to a Second Embodiment 
     The polar code decoding procedure according to the second embodiment is similar to the polar code decoding procedure of the first embodiment in generating the codeword combinations of the blocks with the exception that a number of combinations having the highest posterior probabilities are selected among the codeword combinations between the blocks and the rest are discarded. 
       FIG. 9  is a flowchart illustrating a polar code decoding procedure according to the second embodiment of the present disclosure. 
       FIG. 10  is a diagram illustrating a procedure of decoding a polar code according to the second embodiment of the present disclosure. 
     Referring to  FIGS. 9 and 10 , the receiver (decoder) decodes the received signals by using the SCL decoder at operation  905 . The SCL decoder processes the received signal vectors, i.e., polar code blocks  1010  and  1020 , to generate L codeword candidates per polar code block. The receiver selects/acquires M codeword candidates having the highest posterior probability among the L codeword candidates per block. Here, M is an integer less than L. In the embodiment of  FIG. 10 , L=4 and M=2. In the embodiment of  FIG. 10 , the L codeword candidates acquired from the polar code block  1   1010  include codeword  1   1 , codeword  2   1 , codeword  3   1 , and codeword  4   1 . The receiver selects M codewords (i.e., codeword  1   1 , codeword  2   1 ) among these codeword candidates. Similarly, in the embodiment of  FIG. 10 , the L codeword candidates acquired from the polar code block  2   1020  include codeword  1   2 , codeword  2   2 , codeword  3   2 , and codeword  4   2 . The receiver selects M codewords (i.e., codeword  1   2 , codeword  2   2 ) among these codeword candidates. 
     The receiver sets k to 1 at operation  910 . The receiver generates combinations of codeword candidates of k th  and (k+i) th  blocks at operation  915 . The receiver selects M codeword combinations having the highest posterior probabilities among the generated codeword combinations and rules out the rest at operation  920 . The posterior probability of the codeword combination is identical with the posterior probability of the codeword combination in the first embodiment. Although the same value of M is used for selecting some of the codeword candidates at operation  905  and some of codeword combinations at operation  920 , different values may be used at operations  905  and  920  in an alternative embodiment. 
     The receiver determines whether k+1 is equal to the number of blocks constituting one packet at operation  925 . If it is determined at operation  925  that k+1 is less than the number of blocks constituting one packet, the receiver increments k by 1 at operation  930  and generates new codeword combinations using the previously generated codeword combinations and the codeword candidates (M) of the (k+1) th  block at operation  935 . The receiver selects M codeword combinations having the highest posterior probabilities among the generated codeword combinations at operation  940 . The process is repeated until the k+1 reaches the number of blocks constituting one packet. In the embodiment of  FIG. 10 , K is 2. Accordingly, single stage of k=1 is performed. 
     The receiver combines M codewords (i.e., codeword  1   1 , codeword  2   1 ) acquired from the polar code block  1   1010  and m codewords (i.e., codeword  1   2 , codeword  2   2 ) acquired from the polar code block  2   1020 . As a consequence, four (M 2 ) codeword combinations are acquired as denoted by reference number  1030 . The receiver selects M (i.e., 2) codeword combinations having the highest posterior probabilities among the four codeword combinations  1030 . In the embodiment of  FIG. 10 , the codeword combination of codewords  1   1  and  1   2  and the codeword combination of codewords  1   1  and  2   2  are selected as M codeword combinations having the highest posterior probabilities. 
     The receiver performs error detection on the codeword combinations  1030  using the packet CRC code at operation  945 . The receiver determines whether there is any codeword combination without error at operation  950 . If there is no codeword combination without error, the receiver selects all of the codeword combinations and the procedure goes to operation  960 . If there is any code combination without error, the procedure goes to operation  955 . At operation  955 , the receiver calculates the posterior probability of each codeword combination and selects the codeword combination having the highest posterior probability. 
     According to an alternative embodiment of the present disclosure, if there is no codeword combination without error at operation  950 , the receiver regards this as reception failure and sends the transmitter a NACK. 
     Operations  945 ,  950 ,  955 , and  960  are the process of selecting a codeword combination having the highest posterior probability among the codeword combinations that passed the CRC test without error. Accordingly, it is possible to acquire the same result by performing CRC test on the codeword combinations in the descending order of posterior probability. 
     At operation  960 , the receiver performs decoding on the selected codeword combination and, if the codeword combination is decoded successfully, sends the transmitter an ACK. 
     In the above embodiment, M is set arbitrarily at operation  905 . However, M may be set as follows depending on the embodiment. 
     P (codeword X k ) denotes the posterior probability of the codeword having X th  highest posterior probability in the codeword candidate list corresponding to k th  codeword block. The threshold value is expressed as Th. The size of M may be set differently or identically for the blocks. In the following, the description is directed to the case where the size of M is set differently for the blocks. M is equal to or less than L. However, M has to be set to a value less than L for at least one of the blocks. The receiver may set M to M1 if f(P( 1   t ), . . . , P(L t )) is equal to or greater than Th for the t th  block. In contrast, if f(P( 1   t ), . . . , P(L t ))&lt;Th, the receiver may set M to M2. Here, the function f( ) is the function having the code block of the corresponding block as input. 
     For example, f(P( 1   k ), . . . , P(L k ))=P( 1   k )−P( 2   k ). For example, if the difference between the highest and the second highest posterior probabilities among the posterior probabilities of the L codewords acquired from the k th  block is equal to or greater than the threshold value, M=M1. This means that M for the k th  block is set to M1 if P( 1   k )−P( 2   k )≥Th. In contrast, M for the k th  block may be set to M2=M−1+1 if P( 1   k )−P( 2   k )&lt;Th. The receiver may determine the size of the threshold value depending on the channel condition. 
     The posterior probability of the codeword combination is calculated as the product of the posterior probabilities of the combined codewords as in the first embodiment. 
       FIG. 11  is a diagram illustrating a principle of a decoding procedure according to the second embodiment of the present disclosure. Although L, K, and M are set to small values to simplify the explanation in the embodiment of  FIG. 10 ,  FIG. 11  shows more general case as compared to the embodiment of  FIG. 10 . 
     Referring to  FIG. 11 , a packet consists of K blocks. The individual blocks  1110 ,  1120 , and  1130  are decoded by the SCL decoder. In this embodiment, M codeword combinations are selected among M 2  codeword combinations  1140  generated differently, and the selected codeword combinations are used for combination with the next block. 
     The packet decoding apparatus for performing the procedures of the first and second embodiments may include a communication unit and a control unit. The communication unit may receive/acquire the blocks constituting the packet under the control of the control unit as described in the first and second embodiments. The control unit controls the decoding apparatus to decode the received blocks according to any of the first and second embodiments. 
     The decoding method according to an embodiment of the present disclosure is capable of improving decoding performance of the polar code concatenated in a unit of packets as compared to the method of the related art. 
       FIGS. 12, 13, and 14  are graphs illustrating performance comparison results between the method of the related art and the proposed method according to an embodiment of the present disclosure. 
     In  FIGS. 12 and 13 , the block code length is 512, the coding rate is 0.5, the SCL decoder list size L is 4, and each packet consists of 4 blocks. 
       FIG. 12  shows the performance analysis result on the first embodiment. 
     Referring to  FIG. 12 , the performance comparison is made between the method of the related art and the proposed method according to the first embodiment when M is 2 and 3. In the case of M=2, the proposed method shows superior performance as compared to the method of the related art below the channel quality of 2 dB but inferior performance as compared to the method of the related art over the channel quality of 2 dB. In the case of M=3, the proposed method shows superior performance as compared to the method of the related art over the entire channel quality range. 
       FIG. 13  shows the performance analysis result of the second embodiment. 
     Referring to  FIG. 13 , it shows the performance comparison when the CL is fixed to 3, 6, and 16. In the case of CL=3, the performance of the proposed method is almost equal to that of the method of the related art. In the cases of CL=6 and 16, the proposed method shows the performance improvements of about 0.3 dB and 0.4 dB respectively at the packet error rate of 10 −4 . 
     Referring to  FIG. 14 , it also shows the performance analysis result of the second embodiment. In  FIG. 13 , the block code length is 2048, the coding rate is 0.5, the SCL decoder list size L is 4, and each packet consists of 4 blocks. When CL is fixed to 3, the proposed method of the second embodiment shows the performance gain of about 0.1 dB at the packet error rate of 10 −3  as compared to the method of the related art. 
     As described above, the packet encoding and decoding apparatus and method of the present disclosure is capable of encoding and decoding packets efficiently. 
     It will be understood that each block of the flowchart illustrations and/or block diagrams, and combinations of blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, a special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, implement the functions/acts specified in the flowchart and/or block diagram block or blocks. These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instructions which implement the function/act specified in the flowchart and/or block diagram block or blocks. The computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational operations to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide operations for implementing the functions/acts specified in the flowchart and/or block diagram block or blocks. 
     Furthermore, the respective block diagrams may illustrate parts of modules, segments or codes including at least one or more executable instructions for performing specific logic function(s). Moreover, it should be noted that the functions of the blocks may be performed in different order in several modifications. For example, two successive blocks may be performed at the same time, or may be performed in reverse order according to their functions. 
     The term “module” according to the embodiments of the disclosure, means, but is not limited to, a software or hardware component, such as a Field Programmable Gate Array (FPGA) or an Application Specific Integrated Circuit (ASIC), which performs certain tasks. A module may advantageously be configured to reside on the addressable storage medium and configured to be executed on one or more processors. Thus, a module may include, by way of example, components, such as software components, object-oriented software components, class components and task components, processes, functions, attributes, procedures, subroutines, segments of program code, drivers, firmware, microcode, circuitry, data, databases, data structures, tables, arrays, and variables. The functionality provided for in the components and modules may be combined into fewer components and modules or further separated into additional components and modules. In addition, the components and modules may be implemented such that they execute one or more Central Processing Units (CPUs) in a device or a secure multimedia card. 
       FIG. 15  illustrates a procedure  1500  for decoding a packet comprising a plurality of blocks, according to an embodiment. At block  1510 , the plurality of blocks constituting the packet are acquired. At block  1520 , a plurality of codeword candidates are extracted corresponding to the blocks. At block  1530 , some of the plurality of codeword candidates are selected in a descending order of posterior probability among the plurality of codeword candidates that a corresponding codeword is a correct codeword for decoding a corresponding block. At block  1540 , the selected codeword candidates are combined into a plurality of codeword combinations. At block  1550 , a codeword combination is selected having a highest posterior probability and having passed a cyclic redundancy check (CRC) test without error, among the plurality of codeword combinations. Finally, at block  1560 , the selected codeword combination is decoded. 
       FIG. 16  illustrates an apparatus  1600  for decoding a packet comprising a plurality of blocks. Apparatus  1600  has a communication unit  1620  configured to receive the plurality of blocks constituting the packet  1610 . A control unit  1630  is configured to acquire the plurality of blocks constituting the packet from communication unit  1620 , extract a plurality of codeword candidates corresponding to the blocks, and select some of the plurality of codeword candidates in a descending order of posterior probability among the plurality of codeword candidates that a corresponding codeword is a correct codeword for decoding a corresponding block. Control unit  1630  is further configured to combine the selected codeword candidates into a plurality of codeword combinations, and select a codeword combination having a highest posterior probability and having passed a cyclic redundancy check (CRC) test without error among the plural codeword combinations. Finally, control unit  1630  is further configured to decode the selected codeword combination into a decoded codeword combination  1640 . 
     While the present disclosure has been shown and described with reference to various embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the present disclosure as defined by the appended claims and their equivalents.