Fast Double Error Correcting Code with a Lower Overhead Essay

An Effective Method for Fast Double Error Correcting Code with Lower Overhead

Abstraction

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In digital communicating there are assorted error rectifying codifications and techniques available. In this paper the figure of para spots over the message spots for Hamming, BCH, RS Code, and DEC are examined. This research paper analyzed the operating expense ( para spots ) of assorted systematic individual and dual mistake rectifying codifications. A new codification is proposed for individual and dual mistake rectification, which minimizes the operating expense and encoding/ decryption clip hold.

Keywords:Overhead ( para spots ) Analysis, Encoding/ Decoding Time Delay, Fast dual mistake rectifying codification with lower operating expense.

1. Introduction

The conventional mistake rectifying codifications in digital communicating such as BCH, Reed Solomon codifications have high encoding/ decrypting hold due to linear feedback displacement registry. The whirl codification has high decrypting complexness and uses turn backing that consequences high decoding hold. Overacting codification is easy to implement and has low encoding/ decrypting hold but it is individual mistake rectifying codification. In this paper, an alternate fast, minimum encoding/ decrypting clip hold systematic ( 96, 64 ) dual mistake rectifying codification is proposed. This codification reduces significantly circuit complexness, power ingestion, operating expense, and encoding/ decoding hold. The overhead analysis of some related codifications is presented in subdivision 2. The proposed codification is considered in subdivision 3. The overhead scrutiny of proposed codification is given in subdivision 4.

2. Analysis Parameters

To measure the public presentation of mistake rectifying codification, several public presentation step prosodies such as mistake rectification capableness, encoding decrypting hold and mistake commanding spot overhead are normally used. The analysis of mistake rectifying codifications can be done on the footing of hardware and package public presentation of the codifications.

2.1. Hardware Parameters

This analysis is performed on hardware or simulator. For hardware public presentation of the codification following parametric quantities can be considered:

  1. Operating expense
  2. Probability of Uncorrected Mistakes.
  3. Signal Power
  4. Encoding/ Decoding Time
  5. Encoder/ Decoder Designing Complexity
  6. Number of Hardware Components

The mistake rectifying codifications can be analyzed on the footing of operating expense ( in footings of excess spots ) and error rectification capablenesss of codifications. The term operating expense is used to depict the excess spots. More overhead requires more bandwidth. For illustration, keeping anaudit trailmight consequence in 10 % operating expense, intending that the plan will run 10 % slower when the audit trail is turned on.Programmersoften need to weigh the operating expense of new characteristics before implementing them.

2.2 Software Parameters

When to plan hardware encoder is really complex and dearly-won so package based encoding decryption is used. This uses some encoding decryption algorithm and calculates the para spots ( operating expense ) . A good mistake rectification method has low operating expense and better mistake rectification capablenesss. The public presentation analysis of mistake rectifying codifications can be performed on the footing of package parametric quantities. The analysis parametric quantities are: –

  1. Operating expense
  2. Encoding/Decoding clip hold
  3. Bit Error Ratio ( BER )

2.3 Overhead Analysis

Among above public presentation step parametric quantities, mistake commanding spots operating expense is the of import 1 as it determines the scalability of encoding algorithm and bandwidth. In the communicating for commanding mistakes some excess mistake commanding spots have to be added, this is known as operating expense. Analysis of mistake commanding spots overhead from a theoretical point of position provides a deeper apprehension of advantages, restrictions and trade-offs found in mistake commanding codifications. To measure the mistake commanding spot overhead, the para spots are calculated from codification vector. The analysis is given harmonizing mistake rectification capablenesss in the undermentioned tabular arraies.

Error Correction Code

Code Rate

Operating expense

Single Error Correcting Code [ 1 ]

3/6

50 %

Fast Ultimate ( 8, 4 ) SEC Code [ 2 ]

4/8

50 %

Overacting Code [ 3 ]

( 2m-m-1 ) / ( 2m-1 )

4 / 7

42.85 %

11 / 15

26.66 %

Turbo Code [ 4 ]

11/15

26.66 %

Repeat Code

1/3

200 %

1/5

400 %

1/7

600 %

Table 2.3.1 Overhead Analysis of Single Bit Error Correction Code.

Error Correction Code

Code Rate

Operating expense

BCH Code [ 5 ]

7 / 15

53.33 %

R S Code [ 3 ]

R S ( 255,223 ) [ 6 ]

3 / 7

57.14 %

223/255

DECC [ 7 ]

2 / 8

75 %

Systematic ( 16, 8 ) DEC [ 8 ]

8/16

50 %

DEC BCH [ 9 ]

7 / 15

53.33 %

Turbo Code [ 10 ]

21/31

32.25 %

Table 2.3.2 Overhead Analysis of Double Bit Error Correction Code

3. Proposed Code

The mistake rectifying spots are calculated at the channel encoder. The channel encoder takes input from beginning encoder. The beginning encoder gives a sequence of binary spots ; these binary spots are input for channel encoder. Now cannel encoder divides these spots sequence into thousand size message block. In this algorithm the message block size can be taken as 4, 9, 16, 25, 36, 49, 64, .p2.Where P is any integer figure. This algorithm is more suited for big block size. The para spots overhead is minimized at big message block size but practically it is difficult to plan the encoder. Hence a suited value of K is 64 or 256 can be taken.

In the proposed Single Error Correcting Code para spots are calculated after set uping the message bits into row and column. If spots agreement is done in square format so overhead is minimal. The para spots are calculated for each row and column.

3.1 Encoding Algorithm for individual mistake

( This algorithm calculates the mistake commanding spots ( para spots ) . The message block size is thousand = P2, where P any integer figure. Fiddling stairss are non discussed here merely major stairss given that constitute the encoding procedure. )

while ( having incoming sequence of spots )

divide the incoming sequence of spots into thousand size block

arrange K spots into p* P matrix

calculate para spots PIfor each row ( PI= PIXOR Mij)

calculate para spots PJfor each column ( PJ= PJXOR MJemaah Islamiyah)

terminal while

append PhosphorusIand PJspots at last of message block

Where PIand PJrepresents para spot of IThursdayrow and JThursdaycolumn of matrix.

K message spots

R para spots

entire N spots

3.2 Decoding Algorithm for individual mistake

( This algorithm decodes the receiving codification vector. The codification vector size is n spots, where n = k+r, and k = P2, It finds out the location of mistake and removes the para spots. Fiddling stairss are non given here. )

while ( having transmitted codification vector )

removes the R para spots

take the K information spots from codification vector

arrange K spots into p* P matrix

calculate para spots PIfor each row ( PI= PIXOR Mij)

if calculated para spot lucifers with receive para spot

no mistake in this row

else

there is mistake, set R = row figure

calculate para spots PJfor each column ( PJ= PJXOR MJemaah Islamiyah)

if calculated para spot lucifers with receive para spot

no mistake in this column

else

there is mistake, set degree Celsius = column figure

terminal while

rectify the mistake in matrix corresponding row figure R and column figure degree Celsius

3.3 Double Error Correcting Code

In individual mistake rectification code the para spots are calculated vertically and horizontally. These para spots besides can be calculated by some specific regulations and so these can be appended at last or can be put/ inserted at some ordered location in message spots. In proposed dual mistake rectification codification, para spots are calculated for each row and each column. Other para spots are calculated for each upper left diagonal and upper right diagonal. The encoding algorithm stairss are given below.

3.4 Encoding Algorithm for dual mistake

( This algorithm calculates the mistake commanding spots ( para spots ) . The message block size is thousand = P2, where P any integer figure. Fiddling stairss are non discussed here merely major stairss given that constitute the encoding procedure. )

while ( having incoming sequence of spots )

divide the incoming sequence of spots into thousand size block

arrange K spots into p* P matrix

calculate para spots PIfor each row ( PI= PIXOR Mij)

calculate para spots PJfor each column ( PJ= PJXOR MJemaah Islamiyah)

calculate para spots Pcubic decimeterfor each upper left diagonal

calculate para spots PRfor each upper right diagonal

terminal while

append PhosphorusI, PJ, Pcubic decimeter, and PRspots at last of message block

Where PIand PJrepresents para spot of IThursdayrow and JThursdaycolumn of matrix, and Pcubic decimeter, PRrepresents para spots of left upper and right upper diagonal.

3.5 Decoding Algorithm for Double Error

At the encoder side para spots are appended at message block of size k. The para spots are appended in any order and at decoder side the para spots are removed at that order. After taking the para spots, the message spots are arranged in matrix signifier and once more para spots are calculated and matched with standard para spots.

This algorithm decodes the receiving codification vector. The codification vector size is n spots, where n = k+r, and k = P2, It finds out the location of mistake and removes the para spots. Fiddling stairss are non given here.

while ( having transmitted codification vector )

removes the R para spots

take the K information spots from codification vector

arrange K spots into p* P matrix

calculate para spots PIfor each row ( PI= PIXOR Mij)

if calculated para spot doesn’t lucifer with receive para spot

Message Block

Size K

Code Rate

Overhead in %

64

64/96

33.33 %

100

100 /140

28.57 %

256

256/320

20 %

set RoentgenI= row figure

calculate para spots PJfor each column ( PJ= PJXOR MJemaah Islamiyah)

if calculated para spot doesn’t lucifer with receive para spot

set CJ= column figure

calculate para spots Pcubic decimeterfor each left upper diagonal

if calculated para spot doesn’t lucifer with receive para spot

set Lvitamin D= diagonal figure

calculate para spots Pr for each right upper diagonal

if calculated para spot doesn’t lucifer with receive para spot

set Roentgenvitamin D= diagonal figure

terminal while

rectify the mistake in matrix corresponding row figure R, column figure degree Celsius, left diagonal figure Lvitamin D, and right diagonal figure Rvitamin D.

4 Overhead Analysis for Proposed Algorithm

The value of P is selected harmonizing the designing demand of communicating system. Larger value of p gives little operating expense but greater design complexness. Here overhead is calculated for K = 64, 100, 256. This algorithm calculates 2*p figure of mistake commanding spots, so the per centum operating expense is ( 2*p ) *100 / ( p*p + 2*p ) , where P is figure of row of information matrix.

Message Block

Size K

Code Rate

Overhead in %

64

64/80

20 %

100

100/120

16.66 %

256

256/282

11.34 %

Table 4.1 Overhead Analysis of proposed algorithm for individual spot

Figure 4.1 Comparisons with Other Existing Code

Table 4.2 Overhead Analysis of Proposed Algorithm for dual Mistake

Figure 4.2 Comparisons with Other Existing Code

Decision

Error sensing, rectification commanding mechanisms has been studied and found that it is impossible to rectify all the mistakes but these can be minimized. Still no mistake rectifying codification is available which can rectify all the random mistakes and burst mistakes. The proposed codification is capable to observe and rectify all dual mistake forms. It has minimum overhead with comparing above discussed codification and can be implemented for any application. The proposed codifications do non hold feedback or turn backing so encoding/ decryption clip hold is besides low.

Mentions:

[ 1 ] . Patric R. J. Ostergbrd, Markku K. Kaikkonen, “New Single-Error-Correcting Codes, ” IEEE TRANSACTIONS ON INFORMATlON THEORY, VOL. 42, NO. 4, JULY 1996.

[ 2 ] . Kazeminejad emir & A ; Belhaire, “Fast, Minimal Decoding Complexity, System Level, Binary Systematic ( 41, 32 ) Single-Error-Correcting Codes For On-chip DRAM Applications, ” Proceedings of the 2001 IEEE International Symposium on Defect and Fault Tolerance in VLSI Systems ( DFT’01 ) 1063-6722/01, 2001 IEEE.

[ 3 ] . Sukhjeet Kaur. “Comparative Analysis of Error Correcting Codes for Noisy Channel, ” International Journal of Applied Information Systems ( IJAIS ) – ISSN: 2249-0868 Foundation of Computer Science FCS, New York, USA Volume 3– No.5, July 2012.

[ 4 ] . Hyun Cheol Kim, Pi1 Joong Lee, “Performance of Turbo Codes with a Single-Error Correcting BCH Outer Code, ” ISlT 2000.Sorrento, Italy, June 25-30.2000.

[ 5 ] . MAHMOUD A. SMADI, “PERFORMANCE ANALYSIS OF BPSK SYSTEM WITH HARD DECISION ( 63, 36 ) BCH CODE, ” Journal of Theoretical and Applied Information Technology, pp 177.

[ 6 ] . Sanjeev Kumar, Ragini Gupta, “Performance Comparison of Different Forward Error Correction Coding Techniques for Wireless Communication Systems, ”I S S N: 2 2 2 9 – 4 3 3 3 ( P R I n T ) | I S S N: 0 9 7 6 – 8 4 9 1 ( On cubic decimeter I n vitamin E ) IJCST Vol. 2, Issue 3, September 2011.

[ 7 ] . Joanne Gomes, Dr. B K Mishra, “Performance Analysis of a Novel Double Error Correcting Code for Image Transmission over UWB Channel, ” International Journal of Computer Applications ( 0975 – 8887 ) Volume 10– No.7, November 2010.

[ 8 ] . Ronald Klein and Murali Varanasi, “A Systematic ( 16,8 ) Code for Correcting Double Mistakes, and Detecting Random Triple Errors” , 0-7803-3088-9/96/ $ 4.0001996 IEEE.

[ 9 ] . Paul J. Crepeau,Categorization of Error Locator Polynomials for Double Error Correcting BCH Codes” , IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 8, AUGUST 1998.

[ 10 ] . Hyun Cheol Kim, Pi1 Joong Lee, “Performance of Turbo Codes with a Single-Error Correcting BCH Outer Code” , ISlT 2000.Sorrento, Italy, June 25-30.2000.