Linear Error - Correcting Codes Lecturer : Michel Goemans 1 Linear Error Correcting Codes

We wish to transmit a message (represented in bits) to a destination through a channel reliably. Errors may occur during transmission and we hope to detect and correct these transmitted errors. In order to do so, we must introduce some redundancy in our transmitted message. We saw that Shannon’s Noisy Coding Theorem tells us that if the redundancy is above a certain level (constrained by the noise of the channel), then we can drive the probability of error (detecting the wrong message sent) to zero with increasing blocklength. The proof was based on a random code which is not very practical. For easy encoding and decoding, we need structure in our encoding/decoding functions. We will study a class of codes called linear block codes, because their structure offers several advantages. Linearity will allow an easier analysis of the error correcting ability of a code. Furthermore, the use of matrices to encode/decode messages is much easier than a random codebook, and the code can be concisely described. Error detection means that the code can detect errors but we don’t know where the errors are in a received sequence. Error correction means we know where the errors are and can correct the bit positions in error. If a code can correct up to t errors, then if the sequence contains more than t errors, the decoder may decode to the wrong information sequence. All our messages, codewords, and received messages will be sequences with binary coefficients from the binary field. Let ei be the vector with 1 in the i-th position and zero in all the other positions, so e3 = (00100 . . . 00). For linear codes, encoding is a linear transformation c that maps a message m to a codeword c(m). Block codes means all codewords c(m) have the same length, which we denote as n. Let k be the length of information bits that we encode. Then there are n−k redundancy bits in each codeword, called parity check bits, which are left for error correction. The (n, k) specify the parameters (codeword length and parity check bit length) for a block code. Let c̃ be the received vector or string of n bits. The Hamming distance dH between any two bit strings is the number of positions in which the 2 strings differ. For example, the Hamming distance dH between the codewords c1 = (101101) and c2 = (100110) is 3. Denote d ∗ to be the minimum Hamming distance between any two distinct codewords of a code C as d∗ = dmin = min ci 6=cj dH(ci, cj). (1)