Muhammad Sajjad,Tariq Shah,＊,Mohammad Mazyad Hazzazi,Adel R.Alharbi and Iqtadar Hussain

1Department of Mathematics,Quaid-I-Azam University,Islamabad,45320,Pakistan

2Department of Mathematics,College of Science,King Khalid University,Abha,61413,Saudi Arabia

3College of Computing and Information Technology,University of Tabuk,Tabuk,71491,Saudi Arabia

4Department of Mathematics,Statistics and Physics,Qatar University,Doha,2713,Qatar

Abstract:The decoding algorithm for the correction of errors of arbitrary Mannheim weight has discussed for Lattice constellations and codes from quadratic number fields.Following these lines,the decoding algorithms for the correction of errors oflength cyclic codes (C) over quaternion integers of Quaternion Mannheim (QM) weight one up to two coordinates have considered.In continuation,the case of cyclic codes of lengthsand 2n -1 = p - 2 has studied to improve the error correction efficiency.In this study,we present the decoding of cyclic codes of length n = φ(p) =p - 1 and length 2n - 1 = 2φ(p) - 1 = 2p - 3 (where p is prime integer and φ is Euler phi function) over Hamilton Quaternion integers of Quaternion Mannheim weight for the correction of errors.Furthermore,the error correction capability and code rate tradeoff of these codes are also discussed.Thus,an increase in the length of the cyclic code is achieved along with its better code rate and an adequate error correction capability.

Keywords:Mannheim distance;monoid ring;cyclic codes;parity check matrix extension;syndromes decoding;code rate and error correction capability

1 Introduction

The study of the features of codes and their suitability for various applications is known as coding theory.Data compression,error detection and correction,security,data storage,and data transmission are all performed by codes.Code words are used in some digital communication systems for error correction or detection.Because of this,all code words in a message may have the same pattern of digits.As a result,the message becomes more redundant.As part of the message identification,each code word(without the first)in the message would have a code syndrome.The cyclic code does not utilize these syndromes for error correction.When a message’s code words have been switched in specific sections of the system,simple decoding methods may reveal this.To decide which syndromes are appropriate,random and burst error correction codes are analyzed and compared to one another.

A perfect code is defined as one that achieves a bound (the sphere-packing bound) in a given metric.Perfect codes have long piqued the interest of coding theorists and mathematicians,since they play an essential theoretical and practical role in coding theory.Over finite fields,several perfect codes with regard to the Hamming metric are known,[1-5].In the perspective of Hamming distance,the single error correction of cyclic codes(C)over the ring Zmof integers modulomare defined by Han et al.[6].Then,Tamm et al.[7]established that integer’s cyclic codes for general construction are perfect,and these codes are used for single error correction.

Though,in[8]Huber investigated that cyclic codes over Gaussian integers for two-dimensional signal space are perfect for one Mannheim error,and he further shaded light on the difference in Hamming and Mannheim distances.On the other hand,in[9]Huber substantiated the Mac William’s theorem for the codes having symbols from a finite field with a two-dimensional modulo metric.Neto et al.[10]spoke the cyclic codes over Gaussian integers Z[i]for Mannheim weight one and two,besides he has given a comparison of these cyclic codes with the cyclic codes having symbols from the ring Z[w]of algebraic integers.Nevertheless,Kostadinov[11]derived the bit error probability of the transmitted code word of an integer cyclic code using quadrature amplitude modulation(QAM).For two-dimensional signal patterns like QAM,Severe coding difficulties arise from the algebraic theory of block-cyclic codes over finite fields.

The cyclic,Bose Chaudhuri Hocquenghem(BCH),Srivastava,alternant,and Goppa codes having symbols from a unitary finite commutative ringR,are described by Andrade et al.in[12].Accordingly,for this purpose,they used the factor ring of polynomial ringR[x]in one indeterminate x.?zen et al.[13]has introduced Quaternion Mannheim(QM)distance as a metric and give a decoding procedure of Cyclic codes of lengthover Quaternion integers.Andrade et al.[14,15]has given the modified Berlekamp-Massey decoding algorithm for cyclic,BCH,Goppa,alternan and Srivastava codes designed by the monoid ringanalogue to the codes obtained by polynomial ringR[x].In continuation,Shah et al.[16,17]in the place of a polynomial ring,the construction of cyclic,BCH,Goppa,alternan and Srivastava codes over a finite ring are realized by the monoid ring.

?zen et al.in[18]further contributed some results on the construction of cyclic codes over some finite Quaternion integer rings with respect to the QM distance.However,Güzeltepe et al.[19]has discussed Gaussian,Lipschitz,and Hurwitz weight codes for error correction and revealed that these codes are perfect.A comparison of the code rate,bandwidth,and average energy is also part of the study of[19].Following the cyclic code’s design through monoid ring as described in[14-19]have introduced the decoding ofCover Quaternion integers of lengthfor QM weight two.Moreover,they also discussed the corresponding 2n-1 =p-2 length cyclic codes of thelength cyclic codes for QM weight one and two.

The goal of this work is to demonstrate a decoding procedure for cyclic codes of lengthn=φ(p)=p-1 with QM weight one and two by following the lines drawn in[13,20].In addition,followed monoid ring technique given in[20]for the designing of cyclic codes,the decoding procedure of the cyclic codes of length 2n-1=2φ(p)-1=2p-3 with QM weight one and two is obtained.Thus,a gain in the increase code rate of cyclic codes due to primep,is achieved.

The rest of the paper is laid out as follows:Related work is discussed in Section 2.In Section 3,single and double error-correcting cyclic codes of lengthn=φ(p)=p-1 for QM weight one and two by following techniques in[10,20].In Section 4,the parity check matrix (H) of the cyclic code of lengthn=φ(p)=p-1 is extended to parity check matrix(H)of the cyclic code of length 2n-1=2φ(p)-1=2p-3.Consequently,single and double error-correcting cyclic codes of length 2n-1=2φ(p)-1 for QM weight one and two through monoid rings by following techniques in[20]is obtained.In Section 5,code length and code rates comparison of the cyclic codes for different odd primes.Finally,Section 6,concluded the findings of the study and future directions respectively.

2 Preliminaries

This section provides the key concepts and basic findings that will be used in the study of upcoming sections.First of all,we recall definition of Quaternion integers,QM weight,QM distance and some related results.

2.1 Hamilton Quaternion Integers

2.2 Construction of Cyclic Codes

3 Error Correction of Cyclic Codes of Length n=φ(p)for QM Weights One and Two

This section consists of decoding method forCof lengthn=φ(p)that uses the techniques in[[13],Section 3,and Section 4]and[[17],Section 3,Theorem 2,Lemma 1 and Theorem 3]to rectify single and double errors of QM weight one and two.

3.1 Single Error Correcting Cyclic Codes of QM Weight One

Table 1: ξ =2 is the root of z9+1

3.2 Double Error Correcting Cyclic Codes of QM Weight One

3.3 Single Error Correcting Cyclic Codes of QM Weight Two

3.4 Double Error Correcting Cyclic Codes of QM Weight Two

4 Parity Check Matrix Extensions of n=φ(p)Length to 2n-1=2φ(p)-1 Length Cyclic Codes and Error Correction of these Codes for QM Weight One and Two through Monoid Rings

4.1 Single Error Correcting Cyclic Codes of Length 2n-1=2φ(p)-1 for QM Weight One

ξl 2 Value ξl 2 Value ξl 2 Value(ξ1 2)0 1 (ξ1 2)6 3i+3j+3k (ξ1 2)12 2i+2j+2k(ξ1 2)1 2 (ξ1 2)7 -1+i+j+k (ξ1 2)13 3(ξ1 2)2 -i-j-k (ξ1 2)8 -2+2i+2j+2k (ξ1 2)14 2-i-j-k(ξ1 2)3 -2i-2j-2k (ξ1 2)9 -1 (ξ1 2)15 -3i-3j-3k(ξ1 2)4 -3 (ξ1 2)10 -2 (ξ1 2)16 1-i-j-k(ξ1 2)5 -2+i+j+k (ξ1 2)11 i+j+k (ξ1 2)17 2-2i-2j-2k

4.2 Double Error Correcting Cyclic Codes of Length 2n-1=2φ(p)-1 for QM Weight One

4.3 Single Error Correcting Cyclic Codes of Length 2n-1=2φ(p)-1 for QM Weight Two

4.4 Double Error Correcting Cyclic Codes of Length 2n-1=2φ(p)-1 for QM Weight Two

5 Lengths and Code Rates of Cyclic Codes Comparison

Table 3:Code rate verses different odd primes in[13]and[17]

Table 3:Continued

Figures 1,2,3 and 4:Code Rate of C of Length and 2n-1=p-2 for Primes in[13]and[17]

Table 4:Code rate of cyclic code of lengths n=φ(p)and 2n-1=2φ(p)-1 for primes in proposed study

Figures 5,6,7 and 8:Code Rate of Cyclic Code of Lengths n = φ(p) and 2n-1 = 2φ(p)-1 for Primes in Proposed Study

Figures 9 and 10:Mutual comparison of proposed work with previous existing works

We observed that if the length of the cyclic codes increases due to primep,then the code rate and error correction capability ofCwill be better.

6 Conclusions

The following are the contributions of this study for the efficacy of the cyclic codes over Quaternion integers of QM weight.An effective and consistent modified decoding algorithm for the cyclic codes of lengthsn=φ(p)and 2n-1=2φ(p)-1 to obtain the error correction capability has been furnished.The length of cyclic codes increased due to large primep.For a given primep,a higher code rates for cyclic codes of lengthsn=φ(p)=p-1 and 2n-1=2p-3 is achieved as compared to the code rates of cyclic codes having lengthsand 2n-1=p-2.The error correction capability of the cyclic codes of lengthsn=φ(p)=p-1 and 2n-1 = 2φ(p)-1 = 2p-3 has been improved and it is better than the customary case of the cyclic codes of lengthsand 2n-1=p-2.

Furthermore,the decoding procedure on the base of quaternion integers may be extended to the decoding procedure of octonion integers.

Funding Statement:The authors extend their gratitude to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant number R.G.P.1/85/42.

Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.