Soft Decoding Based on Ordered Subsets of Verification Equations of Turbo-Productive Codes Alexandr Kuznetsov 1[0000-0003-2331-6326], Anastasiia Kiian 1[0000-0003-2110-010X], Kateryna Kuznetsova 1[0000-0002-5605-9293], Tetiana Ivko 1[0000-0003-1772-0074], Oleksii Smirnov 2[0000-0001-9543-874X], Dmytro Prokopovych-Tkachenko 3[0000-0002-6590-3898] 1 V. N. Karazin Kharkiv National University, Svobody sq., 4, Kharkiv, 61022, Ukraine kuznetsov@karazin.ua, nastyak931@gmail.com, kate.kuznetsova.2000@gmail.com, t.ivko@outlook.com 2 Central Ukrainian National Technical University, avenue University, 8, Kropivnitskiy, 25006, Ukraine, dr.smirnovoa@gmail.com 3 University of Customs and Finance, st. Volodymyr Vernadsky, 2/4, Dnipro, 49000, Ukraine, omega2@email.dp.ua Abstract. Methods of soft decoding of cascade code constructions based on the schemes-products of linear block codes (Turbo Product Codes) are considered. An approach is being developed based on the iterative exchange of soft solu- tions between block codes constituting a cascade design. It is shown that a se- quential execution of procedures for the formation of ordered subsets of test equations and the logarithms estimation of a likelihood ratio allows decoding of turbo-productive codes according to the criterion of minimizing the erroneous reception of code symbols. Keywords. Cascade Structures, Turbo Product Codes, Soft Decoding, Verifica- tion Equations, Noise Immunity. 1 Introduction A promising area in the development of noise-resistant coding theory is cascade code structures [1-7, 32-33], methods and algorithms for their decoding with an iterative exchange of soft solutions that allow to provide a required noise immunity of discrete message transmission [8-14]. It should be noted that the implementation complexity of decoding methods based on the use of decision functions increases with length of the code and the correcting capacity [14-17]. Decoding complexity can be reduced by using decision functions defined on a preformed subset of check equations [18-20]. At the same time, this decrease also leads to a decrease in the energy gain [19, 20]. Thus, an actual direction of research is a development (improvement) of decoding methods with soft solutions based on decisive functions, which, without significantly reducing the energy gain from coding, would significantly reduce the complexity of practical implementation. A promising direction in this sense is the formation of or- dered subsets of test equations and decoding methods based on them. 2 Theoretical substantiation of the proposed decoding method The theoretical basis for soft decoding methods is a criterion for testing hypotheses, the mathematical justification for which is based on the total probability formula and the Bayes theorem [18-20]. Suppose that one can make mutually M exclusive assumptions (hypotheses) H1 , H 2 , …, H M about the setting of the experience, and an event A can appear only with one of these hypotheses. Then the probability of an event is calculated by the formula of total probability: P  A   P  H1  P  A H1   P  H 2  P  A H 2   ...  P  H M  P  A H M   M   P  H  P  A H , i 1 i i where P  H i  is the probability of the hypothesis H i ; P  A H i  - conditional prob- ability of an event A with this hypothesis. If prior to the experiment, probabilities of the hypotheses were P  H i  , i  1, 2,..., M and as a result of the experiment an event A occurred, then the a posteriori (experimental, subject to the occurrence of the event A ) hy- potheses probabilities are calculated using the Bayes formula: P  Hi  P  A Hi  P  H i A  M , i  1, 2,..., M .  PH  P A H  i 1 i i The Bayes formula makes it possible to calculate the conditional probabilities of occurrences of the following events, taking into account the posterior probabilities of hypotheses, P  H i A  , i  1, 2,..., M . So, if after the first experiment in which an event A occurred, the next experiment B is performed, in which an event may oc- cur, the conditional probability P  B A  is calculated using the formula of total prob- ability, into which not a priori probabilities P  H i  are substituted, but a posteriori, calculated after the occurrence of the event A , probabilities P  H i A  , i.e. we will receive: M P  B A   P  H A P  B H A , i  1, 2,..., M , i 1 i i where H i A is an event A under the hypothesis H i , P  B H i A  is the conditional probability of co-being B under the hypothesis H i and event A . Suppose now that the demodulator, based on the observation of the received signal and noise interference, estimates which of the possible signals Si  S1 , S2 ,..., S M  (from an ensemble of signals with power M ) was transmitted. You will make mutu- ally exclusive assumptions M (hypotheses) that the corresponding signal Si i  1, 2,..., M has been transmitted,. We calculate the posterior probability of the i hypothesis, subject to admission: S * P  Si  P  S * Si  P  Si S *  M , i  1, 2,..., M , (1)  PS  PS * S  i 1 i i where P  Si  - a priori probability of formation S * of a signal Si by the transmitter; P  S * Si  - conditional probability of reception under the condition that the signal Si is formed by the transmitter. It is usually S * represented as a continuous random variable underlying the hy- pothesis testing criteria. Consider the probability distribution function P  S * : M P  S *   PS  PS * S  . i 1 i i P  S * - is a probability distribution function of the mixture of signal and interfer- ence S * , which gives test statistics in the full signal space S1 , S2 ,..., S M  . In equation (1), the value of the function p  S * is the scaling factor, since the val- ue P  S * is obtained by averaging over the entire space of the signals. Consider a case for two signals. Let binary logic elements 1 and 0 be represented by signals S1  1 and S2  1 . A rigid decision rule, called as a maximum likelihood rule, determines a choice of one of the hypotheses (corresponding to the transmission of signals S1 and S2 , accordingly) based on the comparison of probabilities values P  S *  x S1  and P  S *  x S2  the choice of the larger one. For each data bit trans- mitted, it is decided that the signal S1 was transmitted if S *  x falls on the right side of the decision line (indicated  ), or that the signal S2 was otherwise transmitted. A similar decision rule, known as the maximum a posteriori probability (MAP), can be represented as a minimum error probability rule, taking into account the prior probability of data. In general, the MAP rule is expressed as follows:  S1 , if P  S *  x S1   P  S *  x S2  S , (2)  S2 , if P  S *  x S1   P  S *  x S2  where S - value of the signal corresponding to the decision. Thus, expression (2) establishes the rule for choosing one of the hypotheses corre- sponding to the signals S1 and S2 . Using expression (1), we obtain the equivalent expression:  S1 , if P  S1  P  S * S1   P  S2  P  S * S2  S ,  S2 , if P  S1  P  S * S1   P  S2  P  S * S2  where probability M P  S *   PS  PS * S  i 1 i i in both parts of inequality reduced. Using (2) we introduce a function as a ratio of likelihood functions P  S *  x S1  and P  S *  x S2  : P  S1  P  S * S1  F , (3) P  S2  P  S * S2  then the rule for choosing one of the hypotheses is written as  S , если F  1 S 1 . (4)  S2 , если F  1 Let us translate the expression (3), we get:  P  S1    P  S * S1   ln F  ln    ln  .  P  S2    P  S * S2    Thus, a logarithm of the ratio of likelihood functions ln F is a real representation of the soft solution at the decoder input, with first term on right side of the equality being the logarithm of the relations of a priori probabilities P  S1  and P  S2   P  S1   LS  S1 , S2   ln   ,  P  S2   and the second term is the essence of the logarithm of the posterior probability ratio P  S * S1  and P  S * S2  :  P  S * S1   LDS  S1 , S2   ln    P  S * S2     as a result of channel measurements in the receiver. So, the logarithm of the likelihood function LFS  ln F is rewritten as LFS  S1 , S2   LS  S1 , S2   LDS  S1 , S2  . (5) It should be noted that for AWGN channels, the logarithm of the likelihood func- tion as the result of channel measurements of the received mixture of signal and noise in the receiver will be as follows:  1  1  S * 1  2    exp       P  S * S1     2  2      LDS  S1 , S2   ln    ln    P  S * S2    1  1  S * 1  2     2 exp   2          2 2 1  S * 1  1  S * 1  2        2 S *. 2   2    Considering the ratio 1 2 Eb 2  ,  N0 Eb where - is the ratio of energy of a binary signal Eb to the spectral power density N0 of the noise N 0 , we obtain: Eb LDS  S1 , S2   4  S*, N0 those value of a logarithm of posterior probability ratio P  S * S1  and P  S * S2  , as a result of channel measurements at the receiver, depends exclusively on the signal- to-noise ratio and the value of the received signal and noise mixture S * . In [20], it was shown that for systematic codes, the soft decision at the decoder output (on a logarithmic scale) about received symbol is written in the form of ex- pression LFDK  S1 , S2 , C1 , C2   LFS  S1 , S2   LDK  c1 , c2  , (6) where LDK  С1 , С2  is the logarithm of the likelihood function relation on the received symbol, obtained as a result of decoding. Substituting (5) into (6) we get: LFDK  S1 , S2 , C1 , C2   LS  S1 , S2   LDS  S1 , S2   LDK  c1 , c2  , (7) those the soft decision at the decoder output depends on three values: LS  S1 , S2  - a logarithm of the ratio of the prior probabilities of the signals S1 and S2 ; -a logarithm of the ratio of the posterior probabilities of the signals S1 and S2 (the result of channel measurements) and LDK  С1 , С2  - a logarithm of ratio of the likelihood functions of binary code symbols C1 and C2 as the result of decoding. To get LFDK  S1 , S2 , С1 , С2  , you need to sum up the individual contributions, since all three components are statistically independent [20]. Soft decoder out- put LFDK  S1 , S2 , С1 , С2  is a real number, providing both the hard decision itself and its reliability. The sign LFDK  S1 , S2 , С1 , С2  sets a hard decision, i.e.:  С  1, if LFDK  S1 , S2 , с1 , с2   0 сi   1 , (8) С2  0, if LFDK  S1 , S2 , с1 , с2   0 where сi is the value of the i -th bit corresponding to the taken decision. An eigenvalue LFDK  S1 , S2 , С1 , С2  determines the reliability of the decision. As a rule, ta value LDK  С1 , С2  has the same sign as LFDK  S1 , S2 , С1 , С2  , thus in- creasing the reliability of the decision. For statistically independent values x and y , the sum of two logarithmic likelihood ratios L( x) and L( y ) is determined by the following expression:  e L x   e L y   L  x     L  y   L  x  y   ln   1  e   e    L x L y (9)     1  sgn  L  x    sgn  L  y    min L  x  , L  y  , where function sgn  z  returns a sign of its argument z , and the sign "" is used to denote the sum of data modulo 2 represented by binary digits. The sign    is used to denote the sum of the logarithms of the likelihood functions, which is defined as the logarithm of the likelihood function of the sum modulo 2 of the corresponding argu- ments. An implementation of the turbo decoding procedure involves the use of decoding methods with a soft solution at the input and a soft solution at the output. During the first iteration on such a decoder, the data is considered equally probable, which gives the initial a priori value LS  S1 , S2   0 in equation (7). Channel measurement gives the value LDS  S1 , S2  that is obtained by taking the logarithm of the ratio of the val- ues P  S *  x S1  and P  S *  x S2  for certain values and is the second member of equation (7). The decoder output LDK  С1 , С2  is information derived from the decod- ing process. For iterative decoding, the external likelihood is fed back to the input (of another composite decoder) to update the prior probability of the next iteration infor- mation, i.e. updates a priori probability: LS  S1 , S2   LDK  С1 , С2  . Thus, the decision in the final decoding of each character of the code sequence and information about its reliability depends on the value LFDK  S1 , S2 , С1 , С2  . Based on equation (7), we write the algorithm that gives an estimate of the soft output of the decoder LDK  С1 , С2  and the resulting estimate LFDK  S1 , S2 , С1 , С2  . 1. Install LS  S1 , S2   0 . 2. We decode with the soft solution the first composite code, i.e. find a soft solu- tion LFDK  S1 , S2 , С1 , С2  . 3. Based on equation (7) we calculate LDK  С1 , С2   LFDK  S1 , S2 , С1 , С2   LS  S1 , S2   LDS  S1 , S2  4. For the following composite code install LS  S1 , S2   LDK  С1 , С2  . 5. With a soft solution, we decode the following composite code, i.e. find a soft so- lution LFDK  S1 , S2 , С1 , С2  . 6. For all composite codes, repeat steps 3-5. 7. The result of turbo decoding is a hard decision about a code symbol с by expres- sion (8) based on the soft decision obtained in the last step LFDK  S1 , S2 , С1 , С2  . Thus, as the analysis of above algorithm shows, the main task in implementation of turbo decoding is a development of efficient soft decoding procedures for compos- ite codes, i.e. development of soft decision LDK  С1 , С2  calculation procedures for an iterative exchange procedure in the process of turbo decoding. We study the procedures for finding the soft solution LDK  С1 , С2  at the decoder output, analyze the possible ways to calculate the last term on the right side of equal- ity (7) - the logarithm of the ratio of the likelihood functions of binary code symbols C1 and C2 as a result of decoding. Consider a linear  n, k , d  block code over a finite field GF (2) . A linear code as a subspace GF k (2)  GF n (2) is defined by the generator matrix G , the lines of which form the basis of the linear space GF k (2) . By definition, for each linear code there is an orthogonal completion - a subspace GF n  k (2)  GF n (2) , all elements of which are orthogonal to the elements of GF k (2) . The basis of the linear space GF n  k (2) is giv- en by the check matrix H , and the mutual orthogonality condition implies equal- ity GH T  0 , where by “0” is meant the k  r matrix of zero elements GF (2) . We write the last equality in the form сH T  0 , where с   с0 , с1 ,..., сn 1  is the ar- bitrary code word of the linear block  n, k , d  code under consideration, i.e. c  GF (2) ci   0,1 . k Taking into account the fact that all elements GF n  k (2) can be expressed in terms of a linear combination of rows of a check matrix H , we have сhiT  0 :,  where hi  hi0 , hi1 ,..., hin1  is an arbitrary vector obtained by a linear combination of rows of a matrix H , i  0,1,..., 2n  k  1 . In other words, the last equality holds for all 2 n  k vectors from GF n  k (q ) and we have a system of test equations:  c0 h00  c1h01  ...  c0 h0n1  0;   c0 h10  c1h11  ...  c0 h1n1  0;  (10)  ... c0 h nk  c1h 2nk 1  ...  c0 h 2nk 1  0.   2 10  1  n1 Suppose now that the code word с   с0 , с1 ,..., сn 1  is taken by the criterion of the maximum a posteriori probability, i.e. the values of the log-rhymes of the posterior probabilities P  S * S1  and P  S * S2  :  P  S * S1   LDS (с j )  LDS  S1 , S2   ln    P  S * S2     about each code symbol с j , j  0,1,..., n  1 as a result of channel measurements of the corresponding signals in the receiver. The logarithms of the relations of a priori probabilities P  S1  and P  S2  , corre- sponding to each of the code symbols с j , j  0,1,..., n  1 we denote  P  S1     LS с j  LS  S1 , S2   ln   .  P  S2   Then, taking into account (7) and rule (9) for the i -th checking equation, we have   LDKi c j   LS  c0   LDS  c0   hi     LS  c1   LDS  c1   hi   ...    0 1       j 1        LS c j 1  LDS c j 1 hi    LS c j 1  LDS c j 1 hi   ...    j 1  n 1 if hi j  1;     LS  cn 1   LDS  cn 1   hi   n1  l  0,  LS  cl   LDS  cl   hil (11)  l j      LS  c0   LDS  c0   hi0     LS  c1   LDS  c1   hi1   ...    n 1 if hi j  0,        LS  cn1   LDS  cn1   hin1   LS  cl   LDS  cl   hil l 0 where the summation of "    " and "  " is carried out according to the rule of adding likelihood logarithms, i.e. by expression (9). If we assume that all the estimates LDKi c j   j  0,1,..., n  1 are statistically inde- pendent (for example, if the test equations are mutually orthogonal), then the resulting   estimate LDK c j will be written as: 2nk 1    LDK  c j  , LDK c j  i (12) i0 where the summation is performed according to the usual arithmetic rule of addition of real numbers.   The soft output of the decoder LFDK с j  LFDK  S1 , S2 , С1 , С2  is a real number, and is determined by the expression (7):     LFDK с j  LS с j  LDS с j  LDK c j     2nk 1 (13)    LS с j  LDS с j     LDK  c j . i i0   The sign LFDK с j sets a tough decision according to rule (8):  С1  1, if LFDK с j  0;    сj   С2  0, if LFDK с j  0.   Expressions (11), (12) and (13) define the decisive function based on using loga- rithms of the ratio of likelihood functions of received signals (calculated using a priori and a posteriori probabilities), as well as the logarithm of the ratio of likelihood func- tions of binary code characters as a result of decoding. The corresponding sum (12) defines the decision function based only on the use of the decoding result. Let us analyze the expression (12). Expanding the summation sign according to rule (11), we obtain that expression (12) contains 2 n  k terms, each of which is the result of summation of the n logarithms of the likelihood of code symbols. In turn, the likelihood logarithms of code symbols are the sum of the likelihood logarithms of the received signals (calculated using a priori and a posteriori probabilities). It is obvious that with an increase in the code parameters  n, k , d  , the number of terms increases rapidly and already with the application n  k  32 of the considered approach it be- comes computationally inexpedient. A promising direction in this sense is the devel- opment of a rule for the formation of ordered subsets of check equations and a theo- retical substantiation on their basis of decisive functions for decoding methods with soft solutions. 3 Conclusions As a result of the conducted research, the method of soft decoding of cascade code constructions with iterative exchange of soft solutions was improved which differs from the known methods by the accelerated procedure of selecting test equations with the most reliable symbols, which allows realizing decoding of code words by the criterion of minimizing the erroneous reception of code symbols and speeding up the process of turbo decoding of concatenated codes. The obtained results may be useful in constructing information security code schemes [21-26], for example, as a real alternative to traditional cryptography for post-quantum applications [27]. 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