Upper Bound on the Bit Error Probability of Systematic Binary Linear Codes via Their Weight Spectra

In this paper, upper bound on the probability of maximum a posteriori (MAP) decoding error for systematic binary linear codes over additive white Gaussian noise (AWGN) channels is proposed. *e proposed bound on the bit error probability is derived with the framework of Gallager’s first bounding technique (GFBT), where the Gallager region is defined to be an irregular highdimensional geometry by using a list decoding algorithm. *e proposed bound on the bit error probability requires only the knowledge of weight spectra, which is helpful when the input-output weight enumerating function (IOWEF) is not available. Numerical results show that the proposed bound on the bit error probability matches well with the maximum-likelihood (ML) decoding simulation approach especially in the high signal-to-noise ratio (SNR) region, which is better than the recently proposed Ma bound.


Introduction
Upper bounds on the maximum a posteriori (MAP) decoding error probability, as a key technique for evaluating the performance of the binary linear codes over additive white Gaussian noise (AWGN) channels, bring a profound impact on the reliable transmission of the next-generation mobile communication systems since they can be used to not only predict the performance without resorting to computer simulations but also guide the design of coding [1]. In order to improve the looseness of the union bound in the low signal-to-noise ratio (SNR) region, many improved upper bounds, on the bit error probability [2][3][4][5] and on the frame error probability [2][3][4][6][7][8][9][10][11][12][13][14][15], are proposed. As surveyed in [1], the improved upper bounds on the bit error probability [2][3][4] are based on Gallager's first bounding technique (GFBT): ≤ Pr E b , y ∈ R + Pr y ∉ R , (2) in which E b denotes the event that represents an error in one of the information bits of the decoded codeword, y denotes the received signal vector, and R denotes an arbitrary region around the transmitted signal vector (called the Gallager region). Divsalar [2] chose the region R to be an Euclidean sphere centered at the point along the line connecting the origin to the transmitted signal vector. Sason and Shamai [3] chose the region R to be a circular cone whose central line passes through the origin and the transmitted signal vector. e upper bounds [2,3] on the bit error probability based on equation (2) can be considered to be simply replaced by the weight spectra A d , 1 ≤ d ≤ n in the upper bound on the frame error probability by where A i,d denotes the number of code words of Hamming weight d encoded by information bits of Hamming weight i and k denotes the dimension of the linear code. However, as noted by Zangl and Herzog [4], computing the expression Pr y ∉ R by the factor 1.0 in (2) means that the worst case of k bit errors is assumed if y falls outside the good region R, and then Zangl and Herzog [4] improved the tangentialsphere bound (TSB) on the bit error probability [3] by computing this probability in a more accurate way. e upper bounds on the bit error probability [2][3][4] require the whole input-output weight enumerating function (IOWEF), which can be applied to both systematic codes and nonsystematic codes. e upper bound on the bit error probability by Ma et al. [16] can be evaluated by calculating partial IOWEF with truncated information weight , which holds only for systematic codes. However, for most codes, the IOWEF is usually not computable. In contrast, it is reasonable to assume that the weight spectra A d , 0 ≤ d ≤ n of codes are available, such as the BCH code [17]. In this paper, different from most of the existing bounds, we derive a tighter upper bound on the bit error probability of systematic binary linear codes via their weight spectra.
e main results as well as the structure of this paper are summarized as follows: (1) In Section 2, we present the preliminaries and necessary notation. e conventional union bound and four reported upper bounds based on GFBT are also reviewed in Section 2. (2) In Section 3, in a detailed way, we rederive the recently proposed bound on the bit error probability by Liu [5], in which the union bound is firstly truncated and then amended for the systematic linear codes over AWGN channels. In this paper, the proposed upper bound on the bit error probability is derived in a much more detailed way by considering more information of the Gallager region R and the truncated IOWEF of the code. Finally, with the framework of GFBT, we derive the upper bound on the bit error probability which requires only the knowledge of weight spectra of the code. (3) In Section 4, numerical examples are given to show that the proposed bound is helpful to evaluate the performance of the systematic binary linear codes which can predict the performance of the code in the high-SNR region, avoiding the time-consuming computer simulations. (4) Section 5 concludes this paper.

Preliminaries
Let F 2 � 0, 1 { } be the binary field. Let C[n, k] be a systematic binary linear block code of dimension k and length n with a generator matrix G � [I k , P], where I k is the k × k identity matrix. Let u ∈ F k 2 be the information vector and c ∈ F n 2 be the associated codeword. We have the encoding function as follows: Considering the binary phase shift keying (BPSK) mapping, we have c ⟶ s by s t � 1 − 2c t for 0 ≤ t ≤ n − 1. Suppose that s is transmitted over an AWGN channel. Let y � s + z be the received vector, where z is a vector of independent Gaussian random variables with zero mean and variance σ 2 . We have the decoding function as follows: Without loss of generality, assume that the all-zero codeword c (0) is transmitted. e conventional union bound and four reported upper bounds based on GFBT are also reviewed in the following sections.

Union Bound.
e simplest upper bound on the bit error probability is the union bound: where E i,d b is the event that there exists at least one codeword of Hamming weight d encoded by information bits of Hamming weight i that is nearer than c (0) to y, and Q( � � d √ /σ) is the pairwise error probability with However, the above conventional union bound is loose and even diverges ( ≥1) in the low-SNR region. en, the improved upper bounds on the bit error probability based on GFBT were proposed, such as the Divsalar bound [2], the tangential-sphere bound (TSB) [3], the improved tangential-sphere bound (ITSB) [4], and the Ma bound [16].

e Divsalar Bound.
In 1999, Divsalar derived a simple upper bound [2] on the bit error probability based on GFBT, where the region R is chosen to be an n-dimensional sphere centered at a scaled transmitted signal vector. Both the radius and the center of the sphere can be optimized. Let d min denote the minimum Hamming weight. Taking into account the definition of A * d in (3), we have the Divsalar bound on the bit error probability: where 2 Discrete Dynamics in Nature and Society 2.3. e Tangential-Sphere Bound. In 2000, Sason and Shamai [3] derived the tangential-sphere bound on the bit error probability based on GFBT, where the region R is chosen to be an n-dimensional circular cone whose central line passes through the origin O and the transmitted signal. Let denote the normalized incomplete gamma function. Taking into account the definition of A * d in (3), we have the TSB with a parameter r on the bit error probability: where e parameter r in the TSB can be optimized by a numerical solution of the following equation: where 2.4. e Improved Tangential-Sphere Bound. In 2001, Zangl and Herzog [4] derived the improved tangential-sphere bound on the bit error probability based on GFBT by computing the expression Pr y ∉ R in a more accurate way. We have the ITSB with a parameter r Ψ on the bit error probability: Discrete Dynamics in Nature and Society e parameter r Ψ in the ITSB can be optimized by a numerical solution of the following equation: where where

Upper Bound on the Bit Error Probability Based on GFBT
3.1. e Gallager Region R. We define the region R by the Hamming distance based on a list decoding algorithm which is shown in Figure 1, resulting in an irregular high-dimensional geometry (Algorithm 1). e list decoding algorithm is similar to but different from the algorithm presented in [14]. e list region in [14] is an n-dimensional Hamming sphere with center at the hard decision of the whole received sequence, while the list region here is a k-dimensional Hamming sphere with center at the hard decision of the information part of the received sequence.
e Gallager region R can be defined by

Upper Bound on the Bit Error Probability via IOWEF.
We assume that A i,d ≥ 1 and denote all the code words of Hamming weight d encoded by information bits of Ham- With the framework of GFBT, we have As shown in Figure 1(b), we have As shown in Figure 1(a), we have which means that the decoder outputs at most i + r * erroneous bits. Assuming a binary vector of total length N t passes through a BSC with cross error probability p, we denote B(p, N t , N ℓ , N u ) to be the probability that the number of bit errors occurring ranges from N ℓ to N u , that is,

Theorem 1. We have the upper bound on the bit error probability of systematic binary linear codes under MAP decoding
4 Discrete Dynamics in Nature and Society Proof. Without loss of generality, we denote Notice that a necessary condition for the event E 0⟶1 is that the corresponding input information sequence of the codeword c (1) is in the list L y . Hence, We have Also notice that, for y ∈ R, We have By combining (30) and (32), we can verify that By the union bounds, we have for i 2 ≤ r * − (i/2) from (33).

Corollary 1. e proposed upper bound on the bit error probability ( eorem 1) can improve the conventional union bound on the bit error probability.
Proof. As to the proposed bound ( eorem 1), note that C r * , p b , k, r * + 1, k � 0, by setting we have Since B(p b , k − i, 0, ⌊r * − (i/2)⌋) ≤ 1, the proof is completed.

Corollary 2.
e proposed upper bound on the bit error probability ( eorem 1) can improve the Ma bound on the bit error probability (18).
Proof. Assuming that we know only partial IOWEF with truncated information weight A i,d , 0 ≤ i ≤ T, 0 ≤ d ≤ n − k + T}, the parameter r * in the proposed bound ( eorem 1) is optimized in the interval [0, ⌊T/2⌋]. eorem 1 can be written as Since B(p b , k − i, 0, ⌊r − (i/2)⌋) is the probability that the number of bit errors occurring in a binary vector of total length k − i, when passing through a BSC with cross error probability p b , it ranges from 0 to r − (i/2). en, it can be verified by to complete the proof.
□ e objective of this paper is to derive the upper bound on bit error probability with only knowing of the weight spectrum.

Upper Bound on the Bit Error Probability via Weight
Spectra. In this section, we focus on how to derive the upper bound on the bit error probability via weight spectra. e IOWEF is usually not computable, but the weight spectra A d , 0 ≤ d ≤ n of the code are usually available, such as the BCH code [17]. Let T ≥ 0 be a positive integer that is relatively small. Assuming that we know only the truncated which can be obtained by using a brute-force method and the weight spectrum for 0 ≤ d ≤ n. en, we focus on how to obtain the upper bound on the bit error probability by using the IOWEF {A i,d , 0 ≤ i ≤ T, 0 ≤ d ≤ n − k + T} and the weight spectrum A d , 0 ≤ d ≤ n . We derive the upper bound in the two following cases.
Case 1: if the radius of the Hamming sphere r * ∈ [0, ⌊T/2⌋] in Figure 1, we can get the IOWEF {A i,d , 0 ≤ i ≤ T, 0 ≤ d ≤ n − k + T} by using a brute-force method, and we have (48) Firstly, it is easy to verify that the first term in the righthand side (RHS) of (49) (50) for d ∈ [T + 1, 2r * ] and i is obviously not greater than min d, k for Secondly, it is easy to verify that the second term in the RHS of (49) for d ∈ [T + 1, 2r * + n − k] and i is obviously not greater than min 2r * , k erefore, by combining (53) and (58) with (59). Finally, we have by combining (48) and (59) with (47). en, we have the following theorem.

Theorem 2. We have the upper bound on the bit error probability of systematic binary linear codes via their weight spectra
where Discrete Dynamics in Nature and Society 7 Pr Proof. We can complete the theorem by combining (46)   Assuming that we know only the truncated IOWEF {A i,d , 0 ≤ i ≤ T, 0 ≤ d ≤ n − k + T} and the proof weight spectrum A d , 0 ≤ d ≤ n , eorem 1 can be written as implying that Pr E b ≤ Pr E b 1 . eorem 2 needs a minimization over 0 ≤ r * ≤ k if the optimal parameter r * is in the interval [0, ⌊T/2⌋], and eorem 2 is exactly eorem 1; if r * is in the interval [⌊T/2⌋ + 1, k], eorem 2 is tighter than eorem 1. erefore, we claim that eorem 2 can improve eorem 1 to complete the proof.

Corollary 4.
e proposed upper bound on the bit error probability ( eorem 2) can improve the Ma bound on the bit error probability (18).
Proof. It can be verified by combining Corollaries 2 and 3.

Remark.
e proposed bound ( eorem 2) has a little higher computational loads than the conventional union bound. Firstly, the overhead is caused by recursively computing B(·, ·, ·, ·) and C(·, ·, ·, ·, ·). e probability B(·, ·, ·, ·) and C(·, ·, ·, ·, ·) is the summation with at most k summands, which are independent of the IOWEF and hence can be calculated and stored for use. Secondly, the overhead is caused by minimizing over r * (0 ≤ r * ≤ k). A brute-force method can be implemented by computing the bound for each r * , which can be done recursively.

Numerical Examples
In this section, we need to point out that the weight spectra of the compared BCH codes can be found in [17]. For all the upper bounds on the bit error probability except the Ma bound, we need the whole IOWEF. en, in this paper, the compared bounds are the Ma bound (18) (18). e proposed upper bound is obtained by this truncated IOWEF and the weight spectrum according to (61) in eorem 2 (note that (61) is different from [5], (23)) since eorem 2 here is derived in a much more detailed way when the Hamming weight d ∈ (T, 2r * ]). As pointed out in [18], multidimensional signal processing plays a very important role in effective data analytics and interpretation. In this paper, we tighten the upper bound by analysing the k-dimensional vector. We can see that the proposed bound is tighter than the Ma bound. We can also see that, for the same code length n, the higher the code rate is, the tighter the Ma bound is. e proposed bound is always tighter whatever the code rate is. Figure 4 shows the comparisons between the upper bounds on the bit error probability of the BCH code [255,239], which are also compared with the simulation results under ML decoding. A partial IOWEF {A i,d , 0 ≤ i ≤ T, 0 ≤ d ≤ n − k + T} with T � 5 of the BCH code [255, 239] can be obtained by using a brute-force method. e Ma bound is obtained by this truncated IOWEF according to (18). e proposed upper bound is obtained by this truncated IOWEF and the weight spectrum according to (61). We can see that the proposed bound is tighter than the Ma bound. We can also see that, the proposed bound coincides nicely with the ML decoding results in the high-SNR region when we only e proposed upper bound is obtained by the whole IOWEF and the weight spectrum according to (61). We can see that the proposed bound is tighter than the Ma bound in the low-SNR region when we know the whole IOWEF. We can also see that, the proposed bound and the Ma bound coincide nicely with the ML decoding results in the high-SNR region.

Conclusions
In this paper, upper bound on the bit error probability of systematic binary linear codes under MAP decoding is derived. e proposed bound just requires the weight spectra of the code, which is helpful when the whole IOWEF of the code is not available. e proposed bound ( eorem 2) is proved to be tighter than the recently proposed Ma bound. e numerical results show that the proposed bound on the bit error probability via weight spectra coincides nicely with the ML decoding results in the high-SNR region, which can predict the BER performance without resorting to computer simulations since the simulation is time-consuming in high-SNR region.
Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.