Question

Problem 8: Error Correction & Detection                                                   (6 points)

Assume a 10 by 6 array of data.

1. How many bit errors can a 2D parity Checksum correct?
2. How many can it detect?
3. Compute data sent for a bit stream of 101011 with CRC using a generator 1010.
   1. What CRC bits are appended to the data.
   2. Randomly assume any bit to have been corrupted. Illustrate how the receiver can detect if the data was corrupted.

Cite your reference. You are allowed to use any online calculator for the calculations

[Good formatting, explanations and descriptions required]

Answer 1

8 Soin a. can correct only l-bit 2D pas Parity Transmitted data: llllO000 10101010 illi! Given that the Assume a lo by 6 Array of dala it can detect.2-bit Errors also . All I and 9 bit Errors can be detected l-bit errors, but 1111 0000 10101010 by a 10100101 Received one bit in error : odd pacity 1101011-t 10100101 사 odd b paocity only 2-bit errors can 2D pacity j be in a grid gseid and pocit but 1-bit Exxoys Can be corrected detect only The data bits are grouped in is calleelated How and column. Example foy. Eve vary 1001 10 110010 01 1010 01.11 - 0 000 Oloolo

a How and Column must contain the incorrect bit. pascity violation will be caused by a single bit erron in one row and one column the intersection of that 1001 O ILOO i < does not check oloo Oi! 1 010 1000 1 does not check Conditions for paocity violations caused by a two bit ecolor. ● Two rows and two columns 0 Two rown (when the errors occur in the same column) @ Two columns (when the errors occur in the Same row).

to be halue to multiply by x3 8 Delude 101011000 by 1010. 344410 -34 remainder 04 After divide value: 01000 16. minus the remainder will give the original We subtract the remainder from the © Meassage M(x)= 1010!! = (80)4 (8') +0/X4+1(x3) + day th88 xotx't x²+xS - HX't x34 XS - (854x2+x'+1) CRC polynomial C(X) = 1010 = (X'+x3) =(x*x') the heighest bit in 3 x3 because divisor Polynomial is of degree 3 X3 (x3 + x2 + x + 1) = (x 57 x 370 x H3 + x3) y&£ x+x4x2= 100011000 - 0100010 101011000 — 1010 Remainder i8:100 message -> x + x6+ 8" + x3 101011000 11010 Decimal value Remainder is:100 100 = 0101010100 original as shown below 101011000

Check): widely Ar Error detection technique used Chelk (CRC) Codes interfeted as polynomial Direcvetic. CRC C Cyclic Clyelic Redundancy bit in toslay's computer netesybe is bageol on lydie known as polynomial codes. since it is possible string to be sent as a polynomial whose coefficients core tee Oand l Valles in the Sotring string to ciew the bit string Hee bit on witte Operations K d-bits *r.bits y D: data bite to be sent Rickl bits bit pattelen Maillematical formula : D*9' XOR R

node d-bit piece of data, that the knows that an error has occurred otherwise the Consider the sending hode wants to send to the receiving the sender and receivese must first agree on a(r+1) bit Pat eeu. Known as a generator. which will denotes as G. we will require most significant bit of G. be ali foragåeu data piece of data D. Ylee Sender will choose r additional bits. R and append Hleeni to D such that the resulting d+r bit pattern (Pinterpreted as a binary numbea) binary numbel) is exactly diuisible by G bising modelo-9 Avitelemetic. ve process withe CRC's is Hus simple. checking the recover diuides the dtr received bits by Q if the remainder is nona 2020. Tee relever of Error data is accepted as being correct.

Hlee Valid code words so, thee receiver discoody for Code (0100, ull} minimum distance is 3 = Asume Mie Sender encodes the data word.10 aß 10/01 and sends. it to the deceiver. teis Coolpo ccord will be changed to 01001 If a 3-bit burst error occurs this patter is not one of flee therefore received pattern. Example: for code {ono,od mentimun distance ise no Error poittelee are detected in steel case fleecee fore eroor Patteou come weight 2. oy i Could be detelted Joe Orgoy patléti set belongs to {1110 would be detected 1101 ; 1100. v by this code for code {1111, 1000} minimum distance is 2 therefore Error patteren of weight I could be peetteen Heat belonge to set {1110, 1101, 0111, 10113 detected. The Error 1, 10117.coould be deteced teip code.