DOO HFWXUH Linear block codes Rectangular codes Hamming codes DOO HFWXUHOLG H LQJOHLQNRPPXQLFDWLRQRGHO - PDF document

6.02 Fall 2012 Lecture #4 End-host 6.02 Fall 2012 Digitize (if needed) Original source Source coding Source binary digits (Òmessage bitsÓ) Bit stream Render/display, etc. Receiving app/user Source decoding Bit stream (bit error correction) Recv samples + Demapper Mapper + Xmit samples Bits Signals (Voltages) over physical link (reducing or removing bit errors) computers Bits Embedding for Structural Separation Encode so that the codewords are from Likely error patterns shouldnt transform one codeword 11 00 0 1 10 single-bit error may cause 00 to be 10 (or 01) 110 000 0 1 100 010 111 001 101 011 01 hypercube + mapping 000 010 codewords, we can: patterns of up to t bit errors if and only if correctpatterns of up to t bit errors if and only if patterns of up to tbit errors while correcting all patterns of t) errors e.g.: d=4, t Linear Block Codescodeword via a Sum of any two codewords is codeword necessary and sufficient for code to be linear. , where is the minimum Linear transformation: C is an n-element row vector containing the codeword D is a k-element row vector containing the message Each codeword bit is a specified linear combination of Each codeword is a linear combination of rows of G. Every linear code can be represented by an systematic form --- ordering is not significant, direct inclusion of k message bits in n-bit codeword is. Corresponds to using invertible transformations on rows and permutations on columns of G to get Message bits Parity bits k n n-k Example: Rectangular Parity Codes D1 D2 D3 D4 P3 P4 P1 Idea: start with rectangular checks for each row and column. Single-bit error in errors in a particular row bit that has the error. Parity for each row row #2 and column #2 for row #2 correct no errors is incorrect is incorrect Claim: The min HD of the rectangular code with rows and error correction (SEC) code. D1 D2 D5 D6 P3 P5 P1 P2 D3 D4 D7 D8 D9 D10 D11 D12 P4 P7 Proof: Three cases. differ in 1 row and 1 col parity differ in either 2 rows OR 2 cols (3) Msgs with HD 3 or more P Task: given k-bit message, compute n-bit codeword. We can use standard matrix arithmetic (modulo 2) to do the job. For example, heres how we would describe the (9,4,4) rectangular 1000101010100100111=DDDDDPPPPP001001101123412345000101011    code word \bDxk\tGkxn=Cxn Receiver gets possibly corrupted word, Calculates all the parity bits from the data bits. If no parity errors, return Single row or column parity bit error bits are fine, return them If parity of row are in error, then the data bit in the wrong; flip it and return the All other parity errors are . Return the data as-is, flag an uncorrectable errorŽ Received codewords 0 0 0 2. Decoder action: ________________ 0 0 1 3. Decoder action: ________________ D1 D2 P1 How Many Parity Bits Do Really We Need? We have represent correction, parity bits need to represent two sets of cases: Case 1: No error has occurred (1 possibility) Case 2: Exactly one of the code word bits has an error (inefficient Hamming codes correct single errors with the --- perfect codesŽ (but not best!) Towards More Efficient Codes: subsets, so a single-bit error set of parity check errors. P2 itself had the error! D 1 + D That way, all message bits will be covered with a Index Binary After receiving the possibly corrupted message (use to indicate possibly erroneous symbol), compute a are zero: no errors to figure out correction Index1234567Binary index001010011100101110111(7,4) codeP1P2D2D3D4 errors Write-out the inequality for errors Elementary CombinatoricsIf the ordering of the m selected objects matters, thenIf the ordering of the m selected objects doesnt matter, then the above expression is too large by a Detects single-digit errors, and transpositions MIT OpenCourseWarehttp://ocw.mit.eduFall 201For information about citing these materials or our Terms of Use, visit: .