Related papers: New Constructions for Query-Efficient Locally Deco…
A k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit x_i of the message by querying only k bits of the codeword C(x), even after some constant…
A code $C \colon \{0,1\}^k \to \{0,1\}^n$ is a $q$-locally decodable code ($q$-LDC) if one can recover any chosen bit $b_i$ of the message $b \in \{0,1\}^k$ with good confidence by randomly querying the encoding $x := C(b)$ on at most $q$…
We prove that the blocklength $n$ of a linear $3$-query locally correctable code (LCC) $\mathcal{L} \colon {\mathbb F}^k \to {\mathbb F}^n$ with distance $\delta$ must be at least $n \geq 2^{\Omega\left(\left(\frac{\delta^2 k}{(|{\mathbb…
A locally decodable code encodes n-bit strings x in m-bit codewords C(x), in such a way that one can recover any bit x_i from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2…
We initiate a study of locally decodable codes with randomized encoding. Standard locally decodable codes are error correcting codes with a deterministic encoding function and a randomized decoding function, such that any desired message…
A locally decodable code (LDC) $C \colon \{0,1\}^k \to \{0,1\}^n$ is an error-correcting code that allows one to recover any bit of the original message with good probability while only reading a small number of bits from a corrupted…
A locally decodable code (LDC) C:{0,1}^k -> {0,1}^n is an error correcting code wherein individual bits of the message can be recovered by only querying a few bits of a noisy codeword. LDCs found a myriad of applications both in theory and…
A locally recoverable code of locality $r$ over $\mathbb{F}_{q}$ is a code where every coordinate of a codeword can be recovered using the values of at most $r$ other coordinates of that codeword. Locally recoverable codes are efficient at…
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…
A binary code Enc$:\{0,1\}^k \to \{0,1\}^n$ is $(0.5-\epsilon,L)$-list decodable if for all $w \in \{0,1\}^n$, the set List$(w)$ of all messages $m \in \{0,1\}^k$ such that the relative Hamming distance between Enc$(m)$ and $w$ is at most…
Locally decodable codes (LDCs) are error-correcting codes $C : \Sigma^k \to \Sigma^n$ that admit a local decoding algorithm that recovers each individual bit of the message by querying only a few bits from a noisy codeword. An important…
A code $C \colon \{0,1\}^k \to \{0,1\}^n$ is a $q$-query locally decodable code ($q$-LDC) if one can recover any chosen bit $b_i$ of the message $b \in \{0,1\}^k$ with good confidence by querying a corrupted string $\tilde{x}$ of the…
We study an approximate version of $q$-query LDCs (Locally Decodable Codes) over the real numbers and prove lower bounds on the encoding length of such codes. A $q$-query $(\alpha,\delta)$-approximate LDC is a set $V$ of $n$ points in…
Locally decodable codes (LDC's) are error-correcting codes that allow recovery of individual message indices by accessing only a constant number of codeword indices. For substitution errors, it is evident that LDC's exist -- Hadamard codes…
In this paper we investigate the role of local information in the decoding of the repetition and surface error correction codes for the protection of quantum states. Our key result is an improvement in resource efficiency when local…
A q-query locally testable code (LTC) is an error correcting code that can be tested by a randomized algorithm that reads at most q symbols from the given word. An important question is whether there exist LTCs that have the ccc-property:…
Reed-Muller codes are among the most important classes of locally correctable codes. Currently local decoding of Reed-Muller codes is based on decoding on lines or quadratic curves to recover one single coordinate. To recover multiple…
Locally Decodable Codes (LDCs) are error-correcting codes $C\colon\Sigma^n\rightarrow \Sigma^m,$ encoding \emph{messages} in $\Sigma^n$ to \emph{codewords} in $\Sigma^m$, with super-fast decoding algorithms. They are important mathematical…
We prove that a binary linear code of block length $n$ that is locally correctable with $3$ queries against a fraction $\delta > 0$ of adversarial errors must have dimension at most $O_{\delta}(\log^2 n \cdot \log \log n)$. This is almost…
Locally decodable channel codes form a special class of error-correcting codes with the property that the decoder is able to reconstruct any bit of the input message from querying only a few bits of a noisy codeword. It is well known that…