Cyclic Low Density Parity Check Codes With the Optimum Burst Error Correcting Capability

Sina Vafi

doi:10.1109/access.2020.3032837

Abstract

The paper presents a new scheme of cyclic codes suitable for the correction of burst errors. This is accomplished by the proper definition of their parity check polynomials in which the difference between orders of every two consecutive elements of the utilized polynomial is unique and in order of power of 2. In the proposed polynomials, the number of applied elements is much lower than their orders (or codes' lengths). This leads to represent codes as a class of low-density parity check (LDPC) codes, while they do not have any 4 cycle in their Tanner graphs. Considering the properties of the circulant matrix and structure of the defined polynomials, it is proven that codes have the optimum burst error-correcting capability. This is evident for short and long length codes. Moreover, it is shown that constructed codes can be combined with Fire codes and demonstrate cyclic codes that are applicable for the simultaneous correction of random and burst errors.

Highlights

C YCLIC codes and their shortened shapes named as shortened cyclic codes are the reputable codes for the protection of data against the burst error
The analysis reveals that their burst error-correcting capability is not the same as their burst error correcting limit as the algorithm used for correction of burst errors is not implemented based on the canonical structure of the circulant parity check matrix [10]
This paper presents a method for constructing cyclic low-density parity check (LDPC) codes with the optimum burst error-correcting capability

Summary

INTRODUCTION

C YCLIC codes and their shortened shapes named as shortened cyclic codes are the reputable codes for the protection of data against the burst error. The first part contains the first n−k rows of the matrix, while the remaining rows (or the last k rows) represent the second part of the matrix Considering this structure, a simpler algorithm for determining burst error correcting capability of codes can be formed, when the sum of columns at the last k rows of the matrix is initially calculated. The sum of C8 = [10010110100000000000000000100] with previous columns will generate the zero span with the length of δ − 8 = 10 in the first part This means that the unique maximal zero span is not gained from the first part of the matrix and the burst error-correcting capability is B(Hn) = 8. CYCLIC CODES WITH THE OPTIMUM BURST ERROR CORRECTING CAPABILITY Let p(x) be a polynomial in GF (2), which is defined by m−2 p(x) = 1 +.

BURST ERROR CORRECTING CAPABILITY OF

SPECIAL CASE

APPLICATION OF PROPOSED LDPC CODES FOR

CONCLUSIONS AND FUTURE WORK