Error detection/correction FOUR WEEK PROJECT 1 ITEMS TO BE DISCUSSED 1.0 OVERVIEW OF CODING STRENGTH (3MINS) Weight/distance of binary vectors Error detection and correction Weight distribution Errasure correction 2.0 CYCLIC CODING (4mins) Cyclic codes structure and their binary properties Systematic cyclic encoding and dividing circuits 3.0 (N-K)-STAGE SHIFT REGISTER CODING (3mins) Error detection with an (n-k)-stage shift register

Error detection/correction FOUR WEEK PROJECT CODING STRENGTH (3MINS) WEIGHT/DISTANCE OF BINARY VECTORS Hamming weight, w(U): number of nonzero elements in U. For a binary vector w(U) equals the number of ones in the vector. Hamming distance between two codes U and V, d(U,V): defines the number of elements in which they differ. The minimum distance of a code gives a measure of the codes minimum capability and determines the codes strength.

Error detection/correction FOUR WEEK PROJECT ERROR DETECTION AND CORRECTION (4MINS) ERROR DETECTION AND CORRECTION Task of the decoder after recieving the vector r, is the estimation of the transmitted code Ui. Maximum likelihood algorithm can be used to express the optimal decoder algorithm. Generally, the error correcting capability of a code t, is defined as the maximum number of guaranteed correctable errors per code word and it is related to the minimum distance between two codes vectors. as : t=(dmin-1)/2

Error detection/correction FOUR WEEK PROJECT 4 ERROR DETECTION AND CORRECTION In general, a t-error-correcting (n-k) linear code is capable of correcting a total of 2^(n-k) error patterns. Error detecting-capability, e, of a code related to minimum distance between two vectors may be expressed as: e = dmin-1 If Aj is the number of code vectors of weight j within an (n,k) linear code, then, the numbers Ao, A1,A2,….,An are termed the weight distribution of the code. 1.0 ERROR DETECTION AND CORRECTION (4MINS)

Error detection/correction FOUR WEEK PROJECT CYCLIC CODING (4mins) CYCLIC CODES STRUCTURE/ BINARY PROPERTIES Defn: Subclass of linear block codes, General characteristics Readily implemented with feedback registers Syndrome calculation easily done with feedback shifts Algebraic structure of cyclic codes lends to efficient coding procedures Components of (n, k) linear code vector can be treated as coeffient of a polynomial. Where the polynomial function only serves as a placeholder for the code vector digits.

Error detection/correction FOUR WEEK PROJECT CYCLIC CODING (4mins) SYSTEMATIC CYCLIC ENCODING Cyclic shift of a code vector: given n, and i, the remainder or end round shift could be obtained analytically. Similarly, cyclic code could be obtained using the generator polynomial. Cyclic code using a systematic encoding procedure leads to a reduction in coding complexity. Message digits are utilized as part of the code vector whereby the message digit is shifted into the rightmost k stages of a codeword register, where the parity digits are appended by placing them in the leftmost n-k stages.

Error detection/correction FOUR WEEK PROJECT CYCLIC CODING (4mins) SYSTEMATIC CYCLIC ENCODING Example of cyclic code in systematic form: Cyclic shift of a codeword polynomial and encoding of a message polynomial requires a division of one polynomial by another. What dividing circuit method does is to make such an operation even easier by by employing a feedback shift register. Computes parity bits Computes Message bits

Error detection/correction FOUR WEEK PROJECT (N-K)-STAGE SHIFT REGISTER CODING (3mins) SYSTEMATIC CYCLIC ENCODING WITH AN (N-K)-STAGE SHIFT REGISTER Cyclic coding in systematic form dealt with computation of essentially, parity bits due to division of an upshifted message polynomial by a generator polynomial. Here we deal with upshifting of the message bits by n-k positions which computes only the parity bits. The parity polynomial is the only remainder after division by generator polynomial which can be found at the register after n shifts through the n-k shifts stage feedback.

Error detection/correction FOUR WEEK PROJECT (N-K)-STAGE SHIFT REGISTER CODING (3mins) SYSTEMATIC CYCLIC ENCODING WITH AN (n-k)-STAGE SHIFT REGISTER Example of systematic cyclic encoding with (n-k)-stage shift register