I Introduction
Successivecancellation (SC) decoding is one of the elements constituting the polar code introduced by Arıkan [1]. This paper investigates the error probability of SC decoding for a source code with decoder side information by extending the results in [2, 12] to general linear source codes. It is shown that if for a given encoder there is a decoder such that the block error probability is , then the block error probability of an SC decoder for the same encoder is . Furthermore, we introduce stochastic successivecancellation (SSC) decoding and show that it is equivalent to the constrainedrandomnumber generator introduced in [7]. It is shown that if for a given encoder there is a decoder such that the block error probability is , then the block error probability of an SC decoder for the same encoder is . It is also shown that the error probability of the symbolwise maximum a posteriori decoding of a linear source code and the SSC decoder of the polar source code goes to zero as the block length goes to infinity.
It should be noted that the results of this paper can be applied to the channel coding as introduced in [2, 10, 12]. In particular, the syndrome decoding is the case when a channel is additive, a parity check matrix corresponds to a source encoding function, the syndrome of a channel output corresponds to a codeword of the source code without decoder side information, and the kernel of the parity check matrix forms the channel inputs, that is, the codewords for a channel code.
Throughout the paper, we use the following notations. For random variable
, let be the alphabet of , be the distribution of , and be the conditional distribution of for a given random variable . Let be the conditional entropy of for a given , where we assume that the base of is the cardinality of. A column vector is denoted by a boldface letter
, where its dimension depends on the context. We define , where is the null string when . Let be a support function defined asIi Symbolwise Maximum A Posteriori Decoding
First, we revisit symbolwise maximum a posteriori (SMAP) decoding, which is used for the conventional decoding of a low density parity check code. Although the symbol error rate (the Hamming distance between a source output and its reproduction divided by the block length ) is discussed with symbolwise maximum a posteriori decoding, we focus on the block error probability (an error occurs when a source output and its reproduction are different, that is, the Hamming distance is positive) throughout this paper.
Let be a pair consisting of a source encoder and a decoder with side information. Let be the codeword of a source output . The decoder is constructed by using functions reproducing the th coordinate as
It should be noted that when is memoryless and is a sparse matrix we can use the sumproduct algorithm to obtain an approximation of .
We have the following theorem.
Theorem 1
The error probability of the code is bounded as
where the right hand side of this inequality goes to zero as when .
Proof:
Let be the th coordinate of . Then we have
(1) 
where the first inequality comes from the union bound, the second inequality comes from the fact that the maximum a posteriori decision minimizes the error probability, and the third inequality comes from the fact that implies .
It is known that, when there is an encoding function such that error probability is close to zero for all sufficiently large [5, 11], where we can use one of the following decoders:

the typical set decoder defined as
where
is a conditional typical set,

the maximum a posteriori probability decoder^{1}^{1}1The right hand side of the third equality of (2) might be called the maximumlikelihood decoder. defined as
(2) where the third equality comes from the fact that when and when .
The following sections show upper bounds of the error probability for several decoders in terms of the error probability of a code , where is an arbitrary decoder. It should be noted that we can use one of the decoders mentioned above. We can reduce the effectiveness of the decoders to that of an arbitrary decoder, where ‘effective’ means that the error probability goes to zero as goes to infinity. For example, [9, 10] show that a decoder using a constrainedrandomnumber generator is effective by showing that the maximum a posteriori probability decoder is effective.
Iii Decoding Extended Codeword
Let be an encoder of a source code with decoder side information. Here, we assume that, for a given there is a function and a bijection such that
(3) 
In particular, this condition is satisfied when is a fullrank matrix. We define the bijection as .
Let and be a partition of , that is, they satisfy and . We call and ordered when and . For a vector , define and so that is a symbol in when for every . In the following, we assume that and , where corresponding index sets and may not be ordered in the bijection . We call the extended codeword of . In the following, we denote omitting the dependence on .
Let be a function that reproduces the extended codeword by using the side information. For a codeword and side information , the source decoder with side information is defined as
(4) 
In the context of the polar source codes, corresponds to unfrozen symbols and corresponds to the final step of SC decoding. We have the following lemma for a general case.
Lemma 1
Let and . Then we have
Proof:
We have
(5) 
where the third equality comes from the fact that is bijective, and in the sixth equality we define
(6) 
and use the fact that for all and there is a unique satisfying and .
In the following, we investigate the decoding error probability for an extended codeword.
Iv SuccessiveCancellation Decoding
This section investigates the error probability of the (deterministic) SC decoding. For a source encoder , let , , , and be defined as in the previous section.
For a codeword and side information , the output of an SC decoder is defined recursively as
by using functions defined as
(7) 
which is known as the maximum a posteriori decision rule, where is the conditional probability defined as
(8) 
by using defined by (6).
To simplify the notation, we define when although does not depend on and . We have the following lemma.
Lemma 2
Proof:
As with the proof in [1], we can express the block error events as , where
is an event where the first decision error in SC decoding occurs at stage . The decoding error probability for a extended codeword is evaluated as
(9) 
where the first inequality comes from the union bound, the second equality comes from the fact that when , and the last inequality comes from the fact that implies .
When the index sets and are not ordered like the polar source codes [2, 12], defined by (7) may not use the full information of a codeword . Borrowing words from [1], treats future symbols as random variables rather than as known symbols. In other words, ignores the future symbols in a codeword . This implies that is different from the optimum maximum a posteriori decoder defined as
The following investigates the error probability of the (deterministic) SC decoding by assuming that the index sets and are ordered, that is, and . This implies that for every , defined by (7) uses the full information of a codeword .
Lemma 3
For a source encoder and decoder with side information, let , , , and be as defined in the previous section, where it is assumed that the index sets and are ordered. Then we have
for all .
Proof:
For , let be the th coordinate of the extended codeword of . Then we have the fact that
(10) 
for all satisfying and , where the second equivalence comes from the fact that is bijective, and the third equivalence comes from (3). Then we have
(11) 
where the first inequality comes from Lemma 7 in the Appendix and the fact that , the second inequality comes from the fact that the maximum a posteriori decision rule minimizes the decision error probability, and the last inequality comes from (10).
From Lemmas 1–3 and the fact that , we have the following theorem, which implies that SC decoding is effective when for a given encoding function there is an effective decoding function .
Theorem 2
For a source code with decoder side information, error probability of the (deterministic) SC decoding is bounded as
where the right hand side of this inequality goes to zero as when .
It should be noted again that the index sets and are ordered, while they are not ordered in the original polar source code. In contrast, we can use an arbitrary function that satisfies the assumption and rearrange the index sets and so that they are ordered, while they are fixed in the original polar source code.
V Stochastic SuccessiveCancellation Decoding
This section introduces stochastic successivecancellation (SSC) decoding, which is known as randomized rounding in the context of polar codes.
When , we replace defined in (7) by the stochastic decision rule generating
randomly subject to the probability distribution
for a given . Let be the stochastic decision rule described above. Let be the stochastic decoder by using instead of when . We denote the stochastic decoder corresponding to (4) by . An analysis of the error probability will be presented in the next section.Vi Implementation of SuccessiveCancellation Decoding
In this section, we assume that is a fullrank (sparse) matrix. Without loss of generality, we can assume that the right part of is an invertible matrix. This condition is satisfied for an arbitrary fullrank matrix by using a permutation matrix , where satisfies the condition, and the codeword can be obtained as .
Let be an matrix, where the left part of is an invertible matrix. Then we have the fact that by concatenating row vectors of to , we obtain the invertible matrix , that is, is bijective. By using and , we can construct a successivecancellation decoder that reproduces an extended codeword with and .
Here, let us assume that the left part of is the identity matrix and the right part of is the zero matrix. It should be noted that a similar discussion is possible when the identity matrix is replaced by a permutation matrix.
Since the left part of is the identity matrix, then, for all , the element of is , which is the only positive element in th row of . Then we have the fact that
which implies .
First, we reduce the conditional probability defined by (8). For and , we have
(12) 
where the third equality comes from the fact that and the fourth equality comes from Lemma 8 in the Appendix and the fact that for all . By substituting , we have
(13) 
for and . It should be noted that the right hand side of the second equality appears in the constrainedrandomnumber generation algorithm [7, Eq. (41)]^{2}^{2}2In [7, Eq. (41)], should be replaced by .. This implies that the constrainedrandomnumber generator can be considered as an SSC decoding of the extended codeword specified in the previous section, where we have assumed that this algorithm uses the full information of the codeword for every .
Next, we assume that is memoryless and reduce the condition to improve the algorithm. This idea has already been presented in [8]. Let be the th column vector of . Let be the submatrix of obtained by using and be that obtained by using . At the computation of (13) for , we can assume that has already been determined. Furthermore, we have the fact that the condition is equivalent to . Then, by letting , we can reduce (13) as follows:
(14) 
It should be noted that we can obtain recursively by deleting the leftend column vector of . We can obtain the vector recursively by using the relations
These operations reduce the computational complexity of the algorithm. It should also be noted that the sumproduct algorithm is available for the approximate computation of (14) when is a sparse matrix.
Next, we convert the reproduction of a extended codeword to the reproduction of a source output. When , we have obtained the extended codeword , where . We can reproduce the source output by using the relation , where is the inverse of the concatenation of and . Then we have the relations
from the assumptions of and . Since
we obtain as
where is the inverse of .
Finally, we summarize the decoding algorithm. We assume that is memoryless, is an (sparse) matrix satisfying that is an invertible matrix, and is an matrix satisfying that is an identity matrix.
SC/SSC Decoding Algorithm Using SumProduct Algorithm:

Let and .

Calculate the conditional probability distribution
as(15) by using , , , and , where we define . It should be noted that the sumproduct algorithm can be employed to obtain an approximation of (15).

For the deterministic SC decoding, let be defined as
For the SSC decoding, generate and record a random number subject to the distribution .

Let .

If , then compute , output and terminate.

Let and go to Step 2.
Since the SSC decoder is equivalent to a constrainedrandomnumber generator generating a random sequence subject to the a posteriori probability distribution [7, Theorem 5], we have the following theorem from the fact that the error probability of a stochastic decision with an a posteriori probability distribution is at most twice that of any decision rule [9, Lemma 3].
Theorem 3
For a linear source code with decoder side information, the decoding error of the SSC decoding algorithm is bounded as
where the right hand side of this inequality goes to zero as when .
Vii Analysis When Index Sets Are Not Ordered
In the previous sections, it was assumed that the index sets and corresponding to and are ordered, that is, and . This section investigates the case when they are not ordered. The following lemma asserts that the effectiveness of the decoder is reduced to a condition where the sum of the conditional entropies corresponding to the complement of the codeword goes to zero as .
Lemma 4
Let and be the SC and SSC decoding functions, respectively. Then
Comments
There are no comments yet.