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This paper presents new half rate Quasi Cyclic Low Density Parity Check (QC- LDPC) codes formed on the basis of combinatorial designs. In these codes, circulant matrices of the parity check matrix are formed on the basis of subsets in which the difference between any two elements of a subset is unique with all differences obtained from the same or different subsets. This structure of circulant matrices guarantees non-existence of cycle-4 in the Tanner graph of QC-LDPC codes. First, an irregular code with girth 6 constituted by two rows of circulant matrices is proposed. Then, more criteria will be considered on the structure of subsets with the mentioned feature aiming to represent a new scheme of regular QC-LPDC codes with girth at least 8. From simulations, it is confirmed that codes have similar to or better performance than other well-known half rate codes, while require lower complexity in their design.

Quasi-Cyclic Low Density Parity Check (QC-LDPC) codes are represented as reputable structured-type LDPC codes, which are considered in the current and next generations of broadband transmission and storage systems [^{1}. On the other hand, half rate of these codes with girth 6 and short lengths has been interested in some applications such as multirate transmission systems affected by fading phenomenon [

It is also possible to have half rate of QC-LDPC codes with girth greater than 6 to produce the error floor at lower bit error rates (BERs). These codes are initially designed as a regular code with girth 6 based on approximate cycle extrinsic message degree (ACE) algorithm [

Alternatively, half-rate QC-LDPC codes with the high performance are constructed based on circulant permutation matrices (CPMs). In one method, CPMs with an arbitrary column and row weights are designed based on greatest common divisor (GCD) concept. In addition, a proper masking technique is applied to construct a code with girth at least 8 [

CPM-based parity check matrix of QC-LDPC codes is possibly formed by combination of finite fields and combinatorial designs. In this case, circulant matrices are obtained by combination of two arbitrary subsets of elements from a defined field. Finally, an appropriate masking technique is applied on the obtained CPM-based matrix to construct QC-LDPC codes with girth 8 or higher [

QC-LDPC codes can also be designed on the basis of cyclic difference sets (CDF) in which every specific number of elements defined in the subsets of a group occurs only once [

In this letter, we apply concept of difference sets in constructing two new schemes of QC-LDPC codes. Despite the method presented in [

The rest of paper is organised as follows: Section 2 explains how subsets with unique differences between their elements are formed. Section 3 presents structure of an irregular QC-LDPC code based on subsets defined in section 2. Moreover, it explains how subsets with different lengths are applied to form a regular QC-LDPC code with column weight 3 and girth 8. Section 4 gives simulation results of the newly designed codes and compares their performance with half-rate QC-LDPC codes constructed by other methods and masking techniques. Finally, Section 5 summarises the paper and gives suggestions for the further work.

For given

sequences

1)

where

2) For

for every

Based on this condition, non-zero elements of

where

3) For every

where

For the given

for all

For every

Based on the above constructions, there exists an additive group

such that for all

tisfies this property^{2}.

^{2}It is possible to have greater ν and provide all given conditions.

For example, let

In this section, two new schemes of half rate QC-LDPC codes are presented. In the first method, an irregular code with girth 6 is constructed based on two rows of circulant matrices. In the second method, structure of a regular code with girth 8 formed by more than three rows of circulant matrices is discussed.

Irregular half rate

where

weight 3,

represents the

The above matrix can be viewed as two

Positions of 1 in circulant matrices are based on elements of subsets defined in Equations (1)-(7), where_{i}s. Similarly, there exist

As differences between position of 1s in a circulant matrix with column weight 3 are unique, a cycle-4 will not be obtained from C_{i}s,

As another scheme of half rate QC-LDPC code, the parity check matrix is formed by more than two rows of circulant matrices. This matrix is generally expressed by:

where

with column weights 2 and 1, respectively. Moreover,

Similar to our first scheme, elements of

positions of 1 in

defined in Equations (1)-(7) with

Lemma 1 The parity check matrix given in Equation (9) with

Proof.

shifts of the zeroth row of

column. Similarly, the

and

common 1 and consequently a cycle-4 is formed for the given

By the same argument presented in Lemma 1, it is possible to have other conditions for the existence of a cycle-4, which are dependent on elements of subsets applied in construction of parity check matrix of QC-LDPC code.

Structure of subsets | Cycle-4 condition |
---|---|

As an example,

Lemma 2 In a circulant matrix with length

Proof. By [

At rth row of a circulant matrix with column weight 2 and length

cycle-6 mentioned in above, the first and rth rows with

Proposition 1 The girth of parity check matrix given in (9) with

Proof. In the given

By the same argument in Lemma 2, in the structure of every

The given

The performance of proposed QC-LDPC codes is verified for additive white gaussian noise (AWGN) channel. Codes are modulated by Binary Phase Shift Keying (BPSK) modulation and decoded by Sum Product Algorithm (SPA). Maximum 100 iterations are considered for iterative decoding.

Parity check matrix of the irregular (256, 128) QC-LDPC code is formed by two rows of circulant matrices given in Equation (8). For

girth 6 has close performance to two other (256, 128) codes having girth 8. However,

for

0.25 dB improvement compared to PEG QC-LDPC code [

non-existence of cycle-6 in structure of regular code, which deteriorates effect of harmful trapping sets on the error correcting performance of codes. The results obtained from simulations demonstrate that the error floor of the newly designed codes with girth 8 will be for

The figure also gives the block error rate (BLKER) performance of the constructed QC-LDPC codes. In general, no error floor is observed for

The paper presented new schemes of half rate QC-LDPC codes with girth 6 or 8. They are designed on the basis of difference set property of subsets, which determine structure of constituent circulant matrices. Based on defining new criteria in structure of subsets and proper combination of circulant matrices, regular QC-LDPC codes with girth 8 were obtained. This concluded a high girth code without applying a masking technique. Simulation results confirmed that newly proposed codes have similar performance to other well-known half rate codes, while are designed with the lower complexity. In future work, the performance of constructed codes in the error floor region will be verified by trapping sets analysis and determining their minimum weight.

Vafi, S. and Majid, N.R. (2016) Half Rate Quasi Cyclic Low De- nsity Parity Check Codes Based on Combinatorial Designs. Journal of Computer and Co- mmunications, 4, 39-49. http://dx.doi.org/10.4236/jcc.2016.412002