Locally correctable codes are error-correcting codes that allow a single bit of the codeword to be recovered by looking up a small subset of the coordinates of the received word. In this work, we generalize the concept of traditional locally correcting to locally correcting multiple bits, and present a family of codes which allow multiple bits to be recovered with probability approaching 1 by visiting a small number of locations of the received word, even when a constant fraction of errors exist. Moreover, our codes are of high rate.
Locally correctable codes are error-correcting codes that allow a single bit of the codeword to be recovered by looking up a small subset of the coordinates of the received word. In this work, we generalize the concept of traditional locally correcting to locally correcting multiple bits, and present a family of codes which allow multiple bits to be recovered with probability approaching 1 by visiting a small number of locations of the received word, even when a constant fraction of errors exist. Moreover, our codes are of high rate.
We study a quantum analogue of locally decodable error-correcting codes. A q-query locally decodable quantum code encodes n classical bits in an m-qubit state, in such a way that each of the encoded bits can be recovered with high probability by a measurement on at most q qubits of the quantum code, even if a constant fraction of its qubits have been corrupted adversarially. We show that such a quantum code can be transformed into a classical q-query locally decodable code of the same length that can be decoded well on average (albeit with smaller success probability and noise-tolerance). This shows, roughly speaking, that q-query quantum codes are not significantly better than q-query classical codes, at least for constant or small q.
In the list-of-L decoding of a block code the receiver of a noisy sequence lists L possible transmitted messages, and is in error only if the correct message is not on the list. Consideration is given to (n,e,L) codes, which correct all sets of e or fewer errors in a block of n bits under list-of-L decoding. New geometric relations between the number of errors corrected under list-of-1 decoding and the (larger) number corrected under list-of-L decoding of the same code lead to new lower bounds on the maximum rate of (n,e,L) codes. They show that a jammer who can change a fixed fraction p>
Introduction Error-correcting codes were introduced in the 1940s in order to implement a theorem of Shannon's [260] which guaranteed that virtually error-free communication could be obtained even over a noisy channel. The message to be communicated is first “encoded”, i.e. turned into a codeword, by adding "redundancy". The codeword is then sent through the channel and the received message is “decoded” by the receiver into a message resembling, as closely as possible, the original message. The degree of resemblance will depend on how good the code is in relation to the channel. The usual pictorial representation of this communication link is shown in the schematic diagram Figure 2.1. Here a message is first given by the source to the encoder that turns the message into a codeword, i.e. a string of letters from some alphabet, chosen according to the code used; the encoded message is then sent through the channel, where it may be subjected to noise and hence altered; this possibly-altered message, when it arrives at the decoder belonging to the receiver, is, first of all, equated with the most likely codeword, i.e. the one (should that exist) that, in a probabilistic sense depending on the channel, was probably sent; finally this “most likely” codeword is decoded and the message is passed on to the receiver. Example 2.1.1 Suppose we use an alphabet of just two symbols, 0 and 1, and we have only two messages, for example “no” corresponding to 0, and "yes” corresponding to 1. We wish to send the message “no”, and we add redundancy by simply repeating the message five times. Thus we encode the message as the codeword (00000). The channel interferes, perhaps, with the message and could, for example, change it to (01100). The decoder assesses the message and decides that of the two possible codewords, i.e. (00000) and (11111), the former is the more likely, and hence the message is decoded, correctly, as “no”. Notice that we have made several assumptions here: for example we have assumed that the probability of an error at any position in the word is less than, that each codeword is equally likely to be sent, and that the receiver is aware of the code used.
The binary case was studied in [1], but the method used there doesn't give the tight answer for nonbinary cases and we presented in [2] another method for the corresponding result.Here we formulate the main theorem and prove the auxiliary statements used in [2].During the transmission of q-ary words of length n over the channel at most t errors occur, and the encoder knows the set E of t positions, where these errors are possible.The decoder doesn't know anything about these positions.Let E t = E | E ⊆ {1, 2, . . ., n}, |E| = t be the set of all subsets from {1, 2, . . ., n} of size t and let M be a set of messages (|M| = M ) .A code word x(m, E) depends not only on the message m ∈ M but also on the configuration of possible errors E .So there exists the natural correspondence between the message m ∈ M and the list of code words E∈E t x(m, E) , which we use for the transmission of this message.Thus the code X for the set of messages M represents a collection of M lists E∈E t {x(m, E)}, m ∈ M .Sinde we can use the same word for different configurations, the size of a list can be essentially smaller than the size of the setLet us define the cylinder C(a, A) with the base a = (a, . . ., a n ) and the support A A ⊆ {1, 2, . . ., n} as the set of words (y 1 , . . ., y n ) with y i = a i , if i / ∈ A .It is clear that the size of the cylinder C(a, A) is equal to q |A| and the number of different cylinders with the same support A is equal to q n-|A| .As a result of the transmission of the codeword x(m, E) every word of C x(m, E), E can appear as output of the channel.The code X corrects t localized errors, if the decoder can correctly recover every message m ∈ M .The following condition is necessary and sufficient for it:
We shall not go into details concerning all the technical applications of error-correcting codes. These include satellite pictures, telephone messages via glass fibre using light, compact disc audio system. The idea is as follows. We consider “information” presented as a very long sequence of symbols from a finite set called the “alphabet”. In this course the alphabet will be a finite field IF q . In the sequence each symbol occurs with equal probability. This information is sent to a receiver over a so-called “noisy channel”. In the model that we consider there is a fixed (small) probability p e that a symbol, that is sent over the channel, is changed into one of the other symbols (again, all equally likely). Such an event is called a “symbol-error” and p e is the symbol-error probability. As a result a fraction p e of the transmitted symbols arrives incorrectly at the receiver end of the channel. The aim of coding theory is to lower the probability of error (considerably) at the expense of spending some of the transmission time or energy on redundant symbols. The idea is explained in one sentence as follows. When we read printed text we recognize a printing error in a word because in our vocabulary there is only one word that resembles (is “sufficiently close to”) the printed word.