Post on 02-Feb-2021
Research ArticleA Construction of High Performance Quasicyclic LDPC Codes:A Combinatoric Design Approach
Muhammad Asif ,1 Wuyang Zhou ,1 Muhammad Ajmal ,2
Zain ul Abiden Akhtar ,3 and Nauman Ali Khan 1
1Key Laboratory of Wireless-Optical Communication, University of Science and Technology China, Hefei, 230027, China2School of Mathematical Science, University of Science and Technology China, Hefei, 230027, China3Department of Telecommunication Engineering, οΏ½e Islamia University of Bahawalpur, Bahawalpur, Pakistan
Correspondence should be addressed to Wuyang Zhou; wyzhou@ustc.edu.cn
Received 11 July 2018; Revised 5 November 2018; Accepted 10 December 2018; Published 3 February 2019
Academic Editor: Michael McGuire
Copyright Β© 2019 MuhammadAsif et al.This is an open access article distributed under theCreativeCommonsAttributionLicense,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This correspondence presents a construction of quasicyclic (QC) low-density parity-check (LDPC) codes based on a special typeof combinatorial designs known as block disjoint difference families (BDDFs). The proposed construction of QC-LDPC codesgives parity-check matrices with column weight three and Tanner graphs having a girth lower-bounded by 6. The proposed QC-LDPC codes provide an excellent performance with iterative decoding over an additive white Gaussian-noise (π΄ππΊπ) channel.Performance analysis shows that the proposed short and moderate length QC-LDPC codes perform as well as their competitors inthe lower signal-to-noise ratio (SNR) region but outperform in the higher SNR region. Also, the codes constructed are quasicyclicin nature, so the encoding can be done with simple shift-register circuits with linear complexity.
1. Introduction
Low-density parity-check codes [1] are of vital importancefor many modern communication systems because of theircapacity-approaching performance and low-complexity iter-ative decoding over noisy information channels. LDPC codeswere first discovered by Robert Gallager in the early 1960βsand rediscovered by Mackay [2] in 1990βs. LDPC codesprovidemany advantages over other error correction codes interms of error performance, low-cost encoding anddecoding,and a flexible scale for code length and rate selection.Therefore, LDPC codes have become a focal choice formany advanced communication standards such as Wi-Fi(802.11n/ac/ad), WiMAX (802.16e), and 10 Gigabit Ethernet(802.3an). After several rounds of discussions, LDPC codeshave been determined for 5G communications. LDPC codeshave been adopted by an important scenario of 5G com-munications known as enhancedmobile broadband (eMBB).The most promising error correction codes for 5G commu-nications are polar codes, spatially coupled LDPC codes,binary/nonbinary LDPC codes, block Markov superposition
transmission (BMST), and turbo codes. Recently, in literature[3, 4], low-complexity decoding algorithms have been pre-sented for LDPC codes. Because of these significant efforts,LDPC codes have been adopted for many next-generationcommunication systems.
A binary (π€π, π€π)-regular LDPC code is defined by thenull space of a parity-check matrix π» having constantcolumn-weight π€π and constant row-weight π€π. The nullspace of a parity-checkmatrix having variable column and/orvariable row weights gives an irregular LDPC code. If theparity-check matrix consists of an array circulant permuta-tion matrices of same size over a finite field πΊπΉ(π), the nullspace of this parity-check matrix gives a QC-LDPC codeover πΊπΉ(π) [5β7]. An important constraint on parity-checkmatrix that any two rows or columns of π» can agree in atmost one position, called Row-Column (RC)-constraint.TheRC-constraint on parity-check matrix π» guarantees that theTanner graph of an LDPC code has no length-4 cycles.
Based on the major construction methods, LDPC codesare categorized into two classes: (1) random-like LDPCcodes are designed based on computer search, the most
HindawiWireless Communications and Mobile ComputingVolume 2019, Article ID 7468792, 10 pageshttps://doi.org/10.1155/2019/7468792
http://orcid.org/0000-0002-9699-1675http://orcid.org/0000-0003-2229-2852http://orcid.org/0000-0001-5312-6190http://orcid.org/0000-0002-5661-9107http://orcid.org/0000-0002-7940-1960https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/7468792
2 Wireless Communications and Mobile Computing
well-known random-like constructions are based on PEG[8] and protograph-based methods [9β11]; (2) structuredLDPC codes are constructed based on algebraic techniquessuch as finite fields [5, 6, 12β20], finite geometries [21,22], and combinatorial structures [23β32]. The QC-LDPCcodes, also known as architecture-aware codes, are oneof the most studied LDPC codes because their parity-check matrices have a special structure which facilitatesthe hardware implementations of an encoder and decoder.Compared to random-like LDPC codes, the QC-LDPC codeshave generator matrices which are quasicyclic in nature, soencoding can be done with shift register circuits having linearcomplexity.
Recently, some researchers have focused on a specialclass of regular QC-LDPC codes with parity-check matricescomposed of an array of circulant permutation matrices andproved that the minimum distance of any (π€π, π€π)-regularQC-LDPC code is lower-bounded by π€π + 1 and a girth ofat most 12 [33, 34]. In literature [23β32], QC-LDPC codeshave been constructed based on the different combinatorialstructures. In this correspondence, a constructionmethod forbinary QC-LDPC codes based on block disjoint differencefamilies [35, 36] is presented. The proposed constructionscheme gives length-4 cycles free QC-LDPC codes. Theproposed QC-LDPC codes constructed for short and mod-erate length applications provide excellent error-correctingperformancewith iterative decoding over anπ΄ππΊπ channel.Based on numerical testing, the proposed QC-LDPC codesperform as well as their competitors in the lower SNRregion but outperform in the higher SNR region. Also, thecodes constructed are quasicyclic in nature, so the encodingcan be done with simple shift-register circuits with linearcomplexity.
The remainder of this correspondence is arranged asfollows: in Section 2, the basic concepts about design theoryand cyclic difference families (CDFs) are given.The existenceand construction of block disjoint difference families basedon Skolem and Rosa Triple systems are given in Section 3.Section 4 presents the construction ofQC-LDPC codes basedon BDDFs for V = 1, 3 mod 6. Performance analysis based onnumerical results is presented in Section 5, and the conclusionof this correspondence is presented in Section 6.
2. Basic Concepts and Definitions
In this section, we discuss some basic concepts of design the-ory such as balanced incomplete block design (BIBD), cyclicdifference families, and block disjoint difference families.
Definition 1 (see [37]). A design is a pair (π, π΄), whereπ is aset of elements called points andπ΄ is a collection of nonemptysubsets ofπ called blocks. Let V, k, and π be positive integerssuch that V > π β₯ 2. A (V, π, π)-BIBD is a design (π, π΄) suchthat the following properties hold:
(i) |π| = V,(ii) each block contains exactly π points, and(iii) every pair of points appear in exactly π blocks.
Definition 2 (see [37]). LetπV = {0, 1, . . . , Vβ1} be an additivegroup. The π k-element subsets of πV, π΄ π = {ππ1, ππ2, . . . , πππ},π = 1, 2, . . . , π, ππ1 < ππ2 < β β β < πππ, give a cyclic differencefamily represented as (V, π, π) if all nonzero elements appearπ times among the differences πππ₯ β πππ¦, π = 1, 2, . . . , π, π₯ ΜΈ= π¦,π₯, π¦ = 1, 2, . . . , π. This (V, π, π)-CDF is called a planar CDF ifπ = 1.
In [26β30], (π(πβ1)π+1, π, 1)-CDFs are used to constructQC-LDPC having a girth lower bounded by 6. The existencesof (π(π β 1)π + 1, π, 1)-CDFs are given inTheorem 3.Theorem 3. Under any of the following conditions, a (V, π, 1)-CDF exists:
(i) A (6π+1, 3, 1) cyclic difference family exists for all π β₯1 [38].(ii) A (12π + 1, 4, 1) cyclic difference family exists for all1 β€ π β€ 1000 [39].(iii) A (20π + 1, 5, 1) cyclic difference family exists for 1 β€π β€ 50 and π ΜΈ= 16, 25, 31, 34, 40, 45 [40].In this correspondence, regular QC-LDPC codes of short
and moderate lengths are constructed using a special classof cyclic difference families, called block disjoint differencefamilies.
Definition 4 (see [36]). If πV is an additive group, then afamily of π-tuples of elements from πV is a (V, π, π)-CDF ifthe collection of blocks of π-tuples form a (V, π, π) balancedincomplete block design. If this collection of blocks is disjoint,the family (V, π, π) is known as a block disjoint differencefamily.
In next section, we review some constructions of (V, 3, 1)-BDDFs based on Skolem Triple System and Rosa TripleSystem for V = 1 mod 6 and V = 3 mod 6, respectively.Finally, we will use (V, 3, 1)-BDDFs, V = 1, 3 mod 6, to desinglenght-4 cycles free binary QC-LDPC codes.3. Block Disjoint Difference Families (BDDFs)
3.1. Construction of (v, 3, 1)-BDDFs for v= 1 mod 6. Basedon Skolem Triple System, a class of (V, 3, 1)-CDFs is usedto construct (V, 3, 1)-BDDFs for V = 1 mod 6. First, weconstruct (V, 3, 1)-CDFs based on Skolem Triple Systems,and then a linear translation of (V, 3, 1)-CDFs gives (V, 3, 1)-BDDFs for V = 1 mod 6.Definition 5 (see [37]). A sequence π = (π 1, π 2, . . . , π 2π) of2π elements taken from {1, 2, . . . , π} is known as a Skolemsequence of order π if
(i) every π π β {1, . . . , π} appears exactly twice in π, and(ii) if π π = π π = π for π > π, then π β π = π.Skolem sequences can also be represented as collections
of ordered pairs {(π₯π, π¦π) : 1 β€ π β€ π, π¦π β π₯π = π} withβππ=1{π₯π, π¦π} = {1, 2, . . . , 2π}. Skolem sequences of order πexist if and only ifπ = 0, 1 mod 4. Skolem sequences of orderπ are constructed by a method given in [37].
Wireless Communications and Mobile Computing 3
(i) π = 1; (1, 1)(ii) π = 4; (1, 1, 3, 4, 2, 3, 2, 4)(iii) π = 5; (2, 4, 2, 3, 5, 4, 3, 1, 1, 5)(iv) π = 4πΌ; πΌ β₯ 2
(2πΌ, 4πΌ β 1) , (2πΌ + 1, 6πΌ)(πΌ β 1, 3πΌ) , (πΌ, πΌ + 1)
(4πΌ + π β 1, 8πΌ β π + 1) π = 1, . . . , 2πΌ(πΌ + π + 1, 3πΌ β π) π = 1, . . . , πΌ β 2
(π, 4πΌ β π β 1) π = 1, . . . , πΌ β 2
(1)
(v) π = 4πΌ + 1; πΌ β₯ 2(πΌ + 1, πΌ + 2) , (2πΌ + 1, 6πΌ + 2) , (2πΌ + 2, 4πΌ + 1)
(4πΌ + π + 1, ππΌ β π + 3) , π = 1, . . . , 2πΌ(πΌ + π + 2, 3πΌ β π + 1) , π = 1, . . . , πΌ β 2
(π, 4πΌ β π + 1) , π = 1, . . . , πΌ(2)
Example 6. We construct a Skolem sequence of order 8π = (4, 1, 1, 5, 4, 7, 8, 3, 5, 6, 3, 2, 7, 2, 8, 6) (3)
A Skolem sequence of order π can be used to generate aSteiner Triple System (STS) of order 6π + 1 using followingconstruction:
(i) for each π π = π π β π, form pairs (π, π).(ii) transform each pair into triples (π, π +π, π+π), whereπ π = π π = π.(iii) transform each triple (π, π + π, π + π) into base blocks{0, π, π + π}.(iv) each base block in (iii) is developed by adding 1 underπ6π+1 to generate an STS of order (6π + 1).
Example 7. We construct an STS(25). First, construct askolem sequence of order 4, π = (1, 1, 4, 2, 3, 2, 4, 3). Fromstep (i), we obtain the pairs (1, 2), (4, 6), (5, 8), (3, 7).Based on step (ii), we then convert these pairs intotriples (1, 5, 6), (2, 8, 10), (3, 9, 12), (4, 7, 11). Usingtransformation in step (iii), we construct the sets{0, 1, 6}, {0, 2, 10}, {0, 3, 12}, {0, 4, 11}. We then add 1 toeach of these sets mod 25 to obtain a STS of size 4Γ25 = 100.
From a STS of order (6π + 1), a (6π + 1, 3, 1)-CDF can beobtained by lettingπ΄ π = {0, π, π+π}, for 1 β€ π β€ π, whereπ΄ πβsdenote the base blocks of a (6π+1, 3, 1)-CDF. A constructionof (6π + 1, 3, 1)-BDDFs using a linear translation of (6π +1, 3, 1)-CDFs based on Skolem Triple System can be found in[35].
Theorem 8 (see [35]). οΏ½ere exists a block disjoint (24π +1, 3, 1) difference family for π β₯ 1.
Proof. Beginning with a (24π + 1, 3, 1)-CDF based on aSkolem Triple System of order 4π that does not have disjointblocks, the idea is to linearly translate the blocks such that notwo blocks intersect. This construction requires that π β₯ 3.The cases for smaller values of π are treated separately.
π1: (0, 1, 12π)π1: (0, 4π β 1, 9π β 1)π1: (0, 2π, 10π β 1)π1: (0, 4π, 10π)
π1: (0, 2π β 2π β 1, 7π + π β 1) , 1 β€ π β€ π β 1π1: (0, 2π + 2π, 11π + π β 1) , 1 β€ π β€ π β 1
π1: (0, 2π + 1, 10π + π) , 1 β€ π β€ π β 1β1: (0, 2π, 6π + π) , 1 β€ π β€ π β 1
(4)
By linear translation of above cyclic difference family, wecan obtain the following block disjoint difference family:
π11: (7π β 1, 7π, 19π β 1) , π = 0 mod 2π12: (7π, 7π + 1, 19π) , π = 1 mod 2π11: (2π + 2, 6π + 1, 11π + 1)π11: (0, 2π, 10π β 1)π11: (6π, 10π, 16π)π11: (2π, 2π + 4π β 1, 7π + 3π β 1)π11: (2π + 1, 2π + 4π + 1, 11π + 3π)π11: (17π + 4π, 17π + 2π β 1, π + 3π)β11: (21π + π + 1 + π π, 21π + 3π + 1 + π π, 3π + 2π
+ π π) , π ΜΈβ‘ 0 mod 3β12: (13π + π β ππ + 5, 13π + 3π β ππ + 5, 19π + 2π β ππ
+ 5) π β‘ 0 mod 3 πππ π ΜΈβ‘ π β 1β13: (10π + 13, 12π + 1, 17π + 2) , π β‘ 1 mod 3
(5)
where 1 β€ π β€ π β 1, π π = π mod 2, and ππ is defined asFor π β‘ 0 mod 3: if π ΜΈ= 3π β ππ for all π < π thenππ = 0; otherwise ππ = 4.For π β‘ 1 mod 3: if π ΜΈ= 3π β ππ for all π < π thenππ = 0; otherwise ππ = 2.For π β‘ 2 mod 3: if π ΜΈ= 3π β ππ + 2 for all π < π thenππ = 2; otherwise ππ = 4.
Since by linear translation (and one flip) of cyclic differ-ence family based on Skolem Triple Systems a block disjointdifference family is obtained, clearly, the new set of triplesis also a difference family with disjoint blocks. The lineartranslation from cyclic difference family to block disjoint
4 Wireless Communications and Mobile Computing
Table 1: Linear translation from CDFs to BDDFs for V = 1 mod 6.CDF BDDF Add Commentsπ1 π11 7π β 1 if π β‘ 0 mod 2π1 π12 7π if π β‘ 1 mod 2π1 π11 2π + 2π1 π11 0π1 π11 6tπ1 π11 2π for 1 β€ π β€ π β 1π1 π11 2π + 1 for 1 β€ π β€ π β 1π1 π11 17π + 4π to {0, β(2π + 1), β(10π + π)}for 1 β€ π β€ π β 1β1 β11 21π + π + π π 1 β€ π β€ π β 1 ifπ ΜΈβ‘ 0 mod 3β1 β12 13π + π β ππ + 5 1 β€ π β€ π β 2 ifπ β‘ 0 mod 3β1 β13 10π + 3 if π β‘ 0 mod 3
difference family is summarized in Table 1. In the followingcases for small values of π, 1 β€ π β€ 2:
V = 25 (π = 1): (0,2,9), (6,10,16), (7,8,19), (12,15,20).V = 49 (π = 2): (0,4,19), (2,7,16), (3,9,25), (6,13,23),(8,44,46), (12,20,32), (17,35,37), (21,22,45).
It is relatively easy to verify that the triples are all disjointwhich complete the proof.
3.2. Construction of (v, 3, 1)-BDDFs for v=3 mod 6. Thissubsection gives a construction of (V, 3, 1)-BDDFs for V =3 mod 6. A class of (V, 3, 1)-CDFs based on Rosa TripleSystems is used to construct (V, 3, 1)-BDDFs for V = 3 mod 6.First, we construct (V, 3, 1)-CDFs based on Rosa TripleSystem, then by a linear translation of (V, 3, 1)-CDFs weobtain (V, 3, 1)-BDDFs for V = 3 mod 6.Definition 9 (see [37]). A sequence π =(π 1, . . . , π π, 0, π π+2, . . . , π 2π+1) of 2π + 1 elements takenfrom {1, 2, . . . , π} is said to be a Rosa sequence of order π ifall of the following hold:
(i) every π π β {1, . . . , π} appears exactly twice in π.(ii) if π π = π π = π for π > π, then π β π = π.(iii) a hook or zero is inserted at position π + 1.Rosa sequences can also be expressed as collections of
ordered pairs {(π₯π, π¦π) : 1 β€ π β€ π, π¦π β π₯π = π} withβππ=1{π₯π, π¦π} = {1, 2, . . . , 2π}. Rosa sequences of order π existif and only ifπ = 0, 3 mod 4. A Rosa sequence of orderπ canbe constructed by a method given in [37].
(1) π = 2; {(1, 2), (4, 6)}(2) π = 5; {(1, 5), (2, 7), (3, 4), (8, 10), (9, 12))}
(3) π = 4πΌ; πΌ β₯ 1(2πΌ β 1, 2πΌ) , (3πΌ, 5πΌ + 1)
(3πΌ + 1, 7πΌ + 1) , (6πΌ + 1, 8πΌ + 1)(4πΌ + π + 1, 8πΌ β π + 1) π = 1, . . . , πΌ β 1(5πΌ + π + 1, 7πΌ β π + 1) π = 1, . . . , πΌ β 1
(πΌ + π β 1, 3πΌ β π) π = 1, . . . , πΌ β 1(π, 4πΌ β π + 1) π = 1, . . . , πΌ β 1
(6)
(4) π = 4πΌ β 1; πΌ β₯ 2(6πΌ β 1, 2πΌ) , (5πΌ, 7πΌ + 1)(4πΌ + 1, 6πΌ) , (7πΌ β 1, 7πΌ)
(5πΌ + π, 7πΌ β π β 1) π = 1, . . . , πΌ β 2(4πΌ + π + 1, 8πΌ β π) π = 1, . . . , πΌ β 2
(π, 4πΌ β π) π = 1, . . . , πΌ β 2(7)
Example 10. We construct a Rosa sequence of order 8
π = (3, 1, 1, 3, 6, 7, 5, 8, 0, 4, 6, 5, 7, 4, 2, 8, 2) (8)Rosa sequences of order π can be used to generate SteinerTriple System of order 6π + 3 using following construction[37]:
(i) for each π π = π π β π, form pairs (π, π).(ii) Transform each pair into triples (π, π+π, π+π), whereπ π = π π = π.(iii) Transform each triple (π, π+π, π+π) into sets {0, π, π+π}.(iv) each base block in (iii) is developed by adding 1 underπ6π+3.(v) Add triples of the form {0, 2π + 1, 4π + 2}.(vi) Add 1 to this triple mod 6π+ 3 to generate the STS of
order 6π + 3.
Wireless Communications and Mobile Computing 5
It is important to note that, from step (i) to step (iv),the construction of STS of order 6π + 3 is same as theconstruction of STS of order 6π + 1. But, step (v) and step(vi) are additionally added in the construction of STS oforder 6π + 3. Since, the differences between elements of baseblocks must exist in the difference set {1, 2, . . . , 6π + 2}. Eachbase block of STS of order 6π + 3 in step (iii) covers sixdifferences of the difference set. Then, all the base blocks ofSTS of order 6π + 3 cover 6π differences of the differenceset {1, 2, . . . , 6π + 2}. So, there are two differences which arenot covered by the base blocks in step (iii). To obtain thetwo missing differences, from step (v), a short block of theform {0, 2π + 1, 4π + 2} is added. Finally, from step (vi), theshort block {0, 2π + 1, 4π + 2} is developed by adding 1 oforder V/3 which gives the missing V/3 blocks of STS of order6π + 3.Example 11. We construct an STS of order 27. First, constructa Rosa sequence of order 4, π = (1, 1, 3, 4, 0, 3, 2, 4, 2). Fromstep (i), we obtain the pairs (1, 2), (7, 9), (3, 6), (4, 8).Based on step (ii), we convert these pairs intotriples (1, 5, 6), (2, 11, 13), (3, 7, 10), (4, 8, 12). Usingtransformation in step (iii), we construct the base blocks{0, 1, 6}, {0, 2, 13}, {0, 3, 10}, {0, 4, 12}. From step (iv), eachbase block obtained from step (iii) is developed by adding1 under π27. If we develop all base blocks in (iii) undermod 27, we obtain 108 blocks. But, we know however thatthis is a BIBD and total number of blocks must be equal toπ = V(V β 1)/π(π β 1) = 27(27 β 1)/3(3 β 1) = 117. Also,the differences between elements of four base blocks in step(iii) cover 24 differences of the difference set {1, 2, . . . , 26}and two differences, {9, 18}, are missing. To obtain the twomissing differences, from step (v), a short block of the form{0, 9, 18} is added. Finally, from step (vi), the short block{0, 9, 18} is developed by adding 1 of order 9 which gives themissing 9 blocks of STS of order 27.
From a STS of order (6π + 3), a (6π + 3, 3, 1)-CDF canbe constructed by letting π΄ π = {0, π, π + π}, for 1 β€ π β€ π,where π΄ πβs are called the base blocks of a (6π + 3, 3, 1)-CDF.Theorem 12 gives the construction of (6π+3, 3, 1)-BDDFs bylinear translation of (6π + 3, 3, 1)-CDFs based on Rosa TripleSystem.
Theorem 12 (see [36]). οΏ½ere exists a block disjoint (24π +3, 3, 1) difference family for π β₯ 0.Beginningwith a (24π+3, 3, 1)-CDF based on aRosaTriple
System of order 4π that does not have disjoint blocks, the idea isto linearly translate the blocks such that no two blocks intersect.A detailed construction of (V, 3, 1)-BDDFs based on Rosa TripleSystem can be found in [36].
In next section, we construct two classes of binary QC-LDPC codes based on (V, π, 1)-BDDFs for V = 1, 3 mod 6.4. BDDFs-Based Construction ofQC-LDPC Codes
Consider a parity-check matrix H(1) consisting of an 1 Γ πarray of π Γ π circulant matrices given as follows:
H(1) = [Q1,Q2, . . . ,Qπ] (9)Based on the block disjoint difference families for V =1, 3 mod 6 given in Sections 3.1 and 3.2, we construct a
parity-checkmatrixH(1) where eachQπ, 1 β€ π β€ π, representsa π Γ π circulant permutation matrix whose each row isobtained from the right cyclic shift of the row above it. Thefirst row of Qπ is obtained from one of the π π-element baseblocks of the (V, π, 1)-BDDFs for V = 1, 3 mod 6. The nullspace of H(1) gives a QC-LDPC code of rate π β 1/π and agirth of at least 6.
To make the idea more clear, a detailed construction ofparity-check matrices based on the (V, π, 1)-BDDFs is givenin Example 13, as follows
Example 13. Consider a (25, 3, 1)-BDDF with base blocksπ΄ π = {ππ1, ππ2, . . . , πππ}, π = 1, . . . , 4, and π = 3 for π25. Thebase blocks for a (6 Γ 4 + 1, 3, 1)-BDDF are
π΄1 = {0, 2, 9} ,π΄2 = {6, 10, 16} ,π΄3 = {12, 15, 20} ,π΄4 = {7, 8, 19} .
(10)
Based on above construction, we can construct a matrixB using a (25, 3, 1)-BDDF for V = 1 mod 6:
B = [[[
0 2 9 6 10 16 12 15 20 7 8 199 0 2 16 6 10 20 12 15 19 7 82 9 0 10 16 6 15 20 12 8 19 7
]]]
(11)
Theorem 14. οΏ½e parity-check matrix H(1) based on BDDFsfor V = 1, 3 mod 6 given in (9) has no length-4 cycles.Proof. To prove this theorem, we have to prove that Qπβs, for1 β€ π β€ π, ofH(1) have no length-4 cycles.
Consider a submatrix π given asπ = (π1 π1π2 π2) . (12)
where π1, π2 β Qπ and π1, π2 β Qπ, 1 β€ π, π β€ π. Thesubmatrix π has cycles of length 4 if and only if π1 β π2 =π1βπ2mod V. Due to the property of BDDFs, all the elementsof Qπ and Qπ are distinct. So, both of the differences π1 βπ2 πππ π1 β π2 satisfy the relation that π1 β π2 ΜΈ= π1 β π2.Therefore, thematrixH(1) has no length-4 cycles and providesa girth of at least 6.4.1. A Class of Binary QC-LDPC Codes: Method I. In thissubsection, we give a construction of binary QC-LDPC codesbased on the BDDFs for V = 1, 3 mod 6. Let πΊπΉ(π) be a finitefield. For each nonzero element πΏπ in πΊπΉ(π), 0 β€ π < π β 1,form a (π β 1)-tuple over πΊπΉ(2), uπ(πΏπ) = (π’0, π’1, . . . , π’πβ2),where all the components of uπ are equal to zero except theππ‘β component π’π = 1. Subscript βπβ stands for binary. This
6 Wireless Communications and Mobile Computing
(π β 1)-tuple over πΊπΉ(2) is referred to as the binary location-vector of πΏπ. The binary location-vector of additive identity ofπΊπΉ(π) is an all-zero (π β 1)-tuple, uπ(0) = (0, 0, 0, . . . , 0).
Let πΌ be an element of πΊπΉ(π). The right cyclic-shiftof binary location vector uπ(πΌ) of field element πΌ givesthe binary location vector uπ(πΌπΏ) of field element πΌπΏ. IfπΏ is a primitive element of πΊπΉ(π), then the (π β 1)-tuplesof πΌ, πΏπΌ, πΏ2πΌ, . . . , πΏπβ2πΌ, give a (π β 1) Γ (π β 1) circularpermutation matrixWπ(πΌ) over πΊπΉ(2). The matrixWπ(πΌ) iscalled a (πβ1)-fold binary dispersion ofπΌ overπΊπΉ(2). IfπΌ = 0,then the (π β 1)-fold binary dispersion of 0-elementWπ(0) isa (π β 1) Γ (π β 1) all-zero matrix over πΊπΉ(2).
Next, replacing each element ofH(1) given in (9) by its (πβ1)-fold binary matrix dispersion Wb over πΊπΉ(2). We obtainan π Γ ππ arrayH(1)
πover πΊπΉ(2):
H(1)π =[[[[[[[
W0,0 W0,1 β β β W0,ππβ1W1,0 W1,1 β β β W1,ππβ1... ... d ...Wπβ1,0 Wπβ1,1 β β β Wπβ1,ππβ1
]]]]]]]
(13)
whereWπ,π is an (π β 1) Γ (π β 1) circular permutation matrixover πΊπΉ(2), for 0 β€ π < π and 0 β€ π < ππ. ArrayH(1)
πgives an
π(π β 1) Γ ππ(π β 1)matrix over πΊπΉ(2). Since the matrixH(1)π
satisfies the RC-constraint, so the null space of H(1)π
gives anLDPC code whose Tanner graph has a girth of at least 6.
For any pair of integers π€π and π€π, for 1 β€ π€π β€ π and1 β€ π€π β€ ππ. Let H(1)π (π€π, π€π) be a π€π Γ π€π subarray ofH(1)π
and give a π€π(π β 1) Γ π€π(π β 1) matrix over πΊπΉ(2).The null space ofH(1)
π(π€π, π€π) over πΊπΉ(2) gives a binary QC-
LDPC code πΆ(1)ππ of length π€π(π β 1) with rate at least (π€π βπ€π)/π€π andminimumdistance lower bounded byπ€π+1. SinceH(1)π
(π€π, π€π) satisfies the RC-constraint, the Tanner graph ofπΆ(1)ππ has a girth of at least 6. For different choices of π€π andπ€π, the above construction method gives a class binary QC-LDPC codes for various lengths and rates.
4.2. A Class of Binary QC-LDPC Codes: Method II. In thissubsection, we use the concept of incidence matrices toconstruct QC-LDPC codes based on the (V, 3, 1)-BDDFs. Letπ = πV be a set of V varieties or elements. A design (π, π΅)with π π-subsets of π, π΅1, π΅2, . . . , π΅π, called blocks, is knownas (V, π, π, π, π)-BIBD if the following properties hold: (1) eachelement appears in exactly π blocks; (2) each pair of elementsappears in exactly π blocks; and (3) the block size π is smallcompared to the cardinality of π. A (V, π, π, π, π)-BIBD canalso be described by a V Γ π matrix M = (ππ,π) over πΊπΉ(2)defined by the following rule:
ππ,π = {{{1 if π₯π β π΅π0 if π₯π β π΅π. (14)
where matrixM is called the incidence matrix.The incidencematrix of a (V, π, π, π, π)-BIBD satisfies the following proper-ties: (1) each column ofM contains exactly π 1βs; (2) each row
ofM contains exactly π 1βs; and (3) two distinct rows ofM canagree at most π positions.Example 15. Let (π, π΅) be the following (7, 7, 3, 3, 1)-BIBD:
π = {1, 2, 3, 4, 5, 6, 7} ,and π΅ = {124, 235, 346, 457, 561, 672, 713} . (15)
The incidencematrix of this (7, 7, 3, 3, 1)-BIBD is given asfollows:
M =
[[[[[[[[[[[[[[[
1 0 0 0 1 0 11 1 0 0 0 1 00 1 1 0 0 0 11 0 1 1 0 0 00 1 0 1 1 0 00 0 1 0 1 1 00 0 0 1 0 1 1
]]]]]]]]]]]]]]]
. (16)
It is important to note that each row of M is a right cyclicshift of the previous row and the right cyclic shift of last rowreturns the first row. Also, each column of M is a downwardcyclic shift of a column on its left. Therefore, M is a 7 Γ 7circulant permutation matrix.
Consequently, for a (V, π, π)-BDDF with π = 1, theincidence matrix M satisfies all the required properties of aparity-check matrix. Therefore, the null space of M gives a(π€π, π€π)-LDPC code of length π. Also, the incidencematrixMsatisfies the π πΆ-constraint with π = 1. So, the Tanner graphofM has a girth of at least 6.
Based on (V, π, 1)-BDDFs for V = 1, 3 mod 6, considera parity-check matrix H(2) consisting of an 1 Γ π array ofcirculant matrices given as follows:
H(2) = [M1,M2, . . . ,Mπ] (17)where eachMπ, 1 β€ π β€ π, represents a VΓ π incidence matrixover πΊπΉ(2). Clearly, the matrix H(2) satisfies all the requiredproperties of a parity-checkmatrix.ThematrixH(2) is a VΓππmatrix over πΊπΉ(2) with row and column weights 3π and 3,respectively.The null space ofH(2) gives a binary regular QC-LDPC with minimum distance at least 4, rate lower boundedby (π β 1)/π, and a girth of at least 6.5. Numerical Results
In this section, the error correction performance of twoproposed classes of binary QC-LDPC codes, given in theSections 4.1 and 4.2, is compared with randomly constructedLDPC codes and QC-LDPC codes obtained from designtheoretic techniques. Simulation results are obtained by π΅πiterative decoding with maximum number of iterations equalto 50. Also, Binary-phase-shift-keying(π΅πππΎ) transmission isassumed over an π΄ππΊπ channel.
Firstly, suppose we have a (73, 3, 1)-BDDF for V =1 mod 6 andπΊπΉ(73) is the code construction field. Choosing
Wireless Communications and Mobile Computing 7
1 1.5 2 2.5 3 3.5 4 4.5 5 5.510β8
10β7
10β6
10β5
10β4
10β3
10β2
10β1
Eb/N0 (dB)
BER\
BLER
proposed (2376, 2162) (BER) (2376, 2162) [29] (BER) (2376, 2162) [28] (BER)(2376, 2162) [32] (BER)proposed (2376, 2162) (BLER)
(2376, 2162) [29] (BLER)(2376, 2162) [28] (BLER)(2376, 2162) [32] (BLER)
Figure 1: Error-correcting performance of proposed (2376, 2162)QC-LDPC code, a (2376, 2162)QC-LDPC code based on cyclic differencefamilies [29], a (2376, 2162) QC-LDPC code obtained from cyclic difference families [28], and a (2376, 2162) QC-LDPC code constructedbased on the subsets with distinct differences between the elements [32].
π€π = 3 and π€π = 33, we construct a 3 Γ 33 subarray ofH(1)π
of 72 Γ 72 circular permutation matrices over πΊπΉ(2).Subarray H(1)
π(3, 33) is a 216 Γ 2376 matrix with row-
weight 33 and column-weight 3. The null space ofH(1)π
(3, 33)gives a (2376, 2162) binary QC-LDPC code of rate 0.9110.Assuming π΅πππΎ transmission over π΄ππΊπ channel, the bit-error rate (π΅πΈπ ) and block-error rate (π΅πΏπΈπ ) performance ofproposed code decoded with Sum-product algorithm (πππ΄)are shown in Figure 1. Also shown in Figure 1 are the errorcorrecting performances of (2376, 2162) QC-LDPC codesconstructed from design theoretic techniques in literature[28, 29] and a (2376, 2162)QC-LDPCcode constructed basedon the subsets with distinct differences between the elements[32]. Based on the numerical results, the proposed QC-LDPC codes perform almost the same or better than theircompetitors in the waterfall region but outperform in thehigher SNR region.
Secondly, suppose we have a (81, 3, 1)-BDDF for V =3 mod 6 andπΊπΉ(81) is the code construction field. Choosingπ€π = 3 and π€π = 18, we construct a 3 Γ 18 subarray ofH(1)π
of 80 Γ 80 circular permutation matrices over πΊπΉ(2).Subarray H(1)
π(3, 18) is a 240 Γ 1440 matrix with row-weight
equal to 18, column-weight 3. The null space of H(1)π
(3, 18)gives a (1440, 1202) binary QC-LDPC code of rate 0.8347.The π΅πΈπ and π΅πΏπΈπ performance of this code decoded withπππ΄ is shown in Figure 2. Also shown in Figure 2 are theerror correcting performances of a (1440, 1202) PEG-LDPCcode [8], a (1440, 1202)QC-LDPC code obtained from cyclicdifference families [28], and a (1440, 1202) QC-LDPC codeconstructed from π‘-designs [31]. Based on the simulation
results, the proposed QC-LDPC codes perform as well astheir counterparts in the waterfall region but outperform inthe higher SNR region.
Finally, suppose we have a (133, 3, 1)-BDDF for V = 1mod 6. For π = 20, we have the following parameters:V = 133, π = 2660, π€π = 3, π€π = 60, and π = 1. Basedon this design, the parity-check matrix H(2) consists of anarray of 20 133 Γ 133 circulant matrices over πΊπΉ(2). H(2) isa 133 Γ 2660 matrix over πΊπΉ(2) whose null space gives a(2660, 2532) binary regular QC-LDPC code with rate 0.9518and a girth of at least 6. The π΅πΈπ and π΅πΏπΈπ performanceof this code decoded with πππ΄ is shown in Figure 3. Alsoshown in Figure 3 are the error correcting performances ofa (2660, 2532) PEG-LDPC code [8], and a (2660, 2532) QC-LDPC code constructed based on the subsets with distinctdifferences between the elements [32] and a (2660, 2532)QC-LDPC code obtained from cyclic difference families [28].Based on the simulation results, the proposed QC-LDPCcodes perform as well as their counterparts in the waterfallregion but outperform in the higher SNR region.
6. Conclusion
In this correspondence, two classes of binary QC-LDPCcodes have been constructed based on a special type of com-binatorial designs known as block disjoint difference families(BDDFs). Firstly, binary QC-LDPC codes are constructedusing binary matrix dispersion of finite field elements basedon BDDFs for V = 1, 3 mod 6. Secondly, binary QC-LDPCare constructed based on the incidence matrices obtained
8 Wireless Communications and Mobile Computing
1 1.5 2 2.5 3 3.5 4 4.5 510β8
10β7
10β6
10β5
10β4
10β3
10β2
10β1
100
Eb/N0 (dB)
BER\
BLE
R
proposed (1440, 1202) (BER) (1440, 1202) [28] (BER)(1440, 1202) [8] (BER)(1440, 1202) [31] (BER)
proposed (1440, 1202) (BLER)(1440, 1202) [28] (BLER)(1440, 1202) [8] (BLER)(1440, 1202) [31] (BLER)
Figure 2: Error-correcting performance of proposed (1440, 1202) QC-LDPC code, a (1440, 1202) PEG-LDPC code [8], a (1440, 1202) QC-LDPC code constructed from π‘-designs [31], and a (1440, 1202) QC-LDPC code obtained from cyclic difference families [28].
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 610β8
10β7
10β6
10β5
10β4
10β3
10β2
10β1
Eb/N0 (dB)
BER/
BLE
R
proposed (2660, 2532) (BER)(2660, 2532) [28] (BER)(2660, 2532) [32] (BER)(2660, 2532) [8] (BER)
proposed (2660, 2532) (BLER)(2660, 2532) [28] (BLER)(2660, 2532) [32] (BLER)(2660, 2532) [8] (BLER)
Figure 3: Error-correcting performance of proposed (2660, 2532) QC-LDPC code, a (2660, 2532) PEG-LDPC code [8], a (2660, 2532) QC-LDPC code constructed constructed based on the subsets with distinct differences between the elements [32], and a (2660, 2532) QC-LDPCcode obtained from cyclic difference families [28].
from (V, 3, 1)-BDDFs for V = 1, 3 mod 6. The proposed QC-LDPC codes have parity-check matrices with column-weightthree and their Tanner graphs provide a girth of at least 6.Also, the proposed QC-LDPC codes provide an excellenterror performance with iterative decoding over an π΄ππΊπchannel. Based on the simulation results, the performance
analysis shows that the proposed QC-LDPC codes of shortand moderate length perform as well as their competitors inlower SNR region but outperform in the higher SNR region.Also, the codes constructed are quasicyclic in nature, so theencoding can be done with simple shift-register circuits withlinear complexity.
Wireless Communications and Mobile Computing 9
Data Availability
No data were used to support this study.
Disclosure
The current address of Muhammad Asif, Wuyang Zhou,Muhammad Ajmal, and Nauman Ali Khan is No. 96, JinZhaiRoad Baohe District, Hefei, Anhui, 230026, PR China.
Conflicts of Interest
The authors declare that they have no conflicts of interestregarding the publication this work.
Authorsβ Contributions
Muhammad Asif, Wuyang Zhou, Muhammad Ajmal, Zainul Abiden Akhtar, and Nauman Ali Khan conceived anddesigned this research work; Muhammad Asif and WuyangZhou participated in construction and performance analysisof this work. Muhammad Ajmal and Zain ul Abiden Akhtarparticipated in numerical analysis of this work; MuhammadAsif wrote the paper and Nauman Ali Khan technicallyreviewed the paper.
Acknowledgments
This work was partially supported by Natural Science Foun-dation of China under Grant number: 61461136002, KeyProgram of National Natural Science Foundation of Chinaunder Grant number: 61631018, and Fundamental ResearchFunds for the Central Universities and Huawei InnovationResearch Program. Author Muhammad Asif acknowledgesthe support of the Chinese Academy of Sciences (CAS)and TWAS for his PhD studies at the University of Scienceand Technology, China, as a 2016 CAS-TWAS PresidentβsFellowship Awardee (CAS-TWAS No. 2016-48).
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